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For nonnegative integers h , g with h ≥ 2 , a set A of nonnegative integers is defined as a B h [ g ] sequence if, for every nonnegative integer n , the number of representations of n with the form n = a 1 + a 2 + ⋯ + a h is no larger than g , where a 1 ≤ ⋯ ≤ a h and a i ∈ A for i = 1 , 2 , ⋯ , h. Let Z be the set of integers and N be the set of positive integers. In 2013, by introducing the method of Inserting Zeros Transformation , Cilleruelo and Nathanson obtained the following result: let f : Z → N ⋃ { 0 , ∞ } be any function such that lim inf | n | → ∞ f (n) ≥ g and let B be any B h [ g ] sequence. Then, for any decreasing function ϵ (x) → 0 as x → ∞ , there exists a sequence A of integers such that r A , h (n) = f (n) for all n ∈ Z and A (x) ≫ B (x ϵ (x)). Recently, Nathanson further considered Sidon sets for linear forms. In this paper, we apply the Inserting Zeros Transformation into Sidon sets for linear forms and generalize the above result related to the inverse problem of representation functions. For a video summary of this paper, please visit https://youtu.be/KRtew2rLbXY. [ABSTRACT FROM AUTHOR]

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ , where μ ≥ C log ⁡ λ , such that intervals [ λ , λ + μ ] do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in R 2 that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in R 2. Specifically, we establish the existence of annuli { x ∈ R 2 : λ ≤ | x | 2 ≤ λ + κ } with arbitrarily large λ and κ ≥ C λ s for 0 < s < 1 4 , satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold s = 1 4. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in R 3. [ABSTRACT FROM AUTHOR]

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

We say a class C of graphs is clean if for every positive integer t there exists a positive integer w (t) such that every graph in C with treewidth more than w (t) contains an induced subgraph isomorphic to one of the following: the complete graph K t , the complete bipartite graph K t , t , a subdivision of the (t × t) -wall or the line graph of a subdivision of the (t × t) -wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H) is clean if and only if H is a forest whose components are subdivided stars. Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η , a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G. This is of independent interest, and will be used in subsequent papers in this series. [ABSTRACT FROM AUTHOR]

The Gyárfás-Sumner conjecture says that for every forest H and every integer k , if G is H -free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H , if G is H -free and does not contain K t , t as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H , the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer d ≥ 2 , if G is H -free and does not contain as a subgraph the complete d -partite graph with parts of cardinality t , then its chromatic number is at most polynomial in t. [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime, and R = G R (p 2 , p 2 r) is a Galois ring of characteristic p 2 , whose cardinality is p 2 r. For any subset D ⊆ R , let N (D , n , b) and N f (D , n , b) , respectively, be the number of n -subsets, denoted by S , in D such that ∑ x ∈ S x = b and ∑ x ∈ S f (x) = b , where f (x) ∈ R [ x ] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N (D , n , b) and N f (D , n , b) under some certain constraints of D and f (x). The special cases D = { a k : a ∈ R ⁎ } and f (x) = x k were studied by the authors in a previous paper, where R ⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings. [ABSTRACT FROM AUTHOR]

For positive integers u and v , let L u = [ 1 0 u 1 ] and R v = [ 1 v 0 1 ]. Let G u , v be the group generated by L u and R v. In a previous paper, the authors determined a characterization of matrices M = [ a b c d ] in G u , v when u , v ≥ 3 in terms of the short continued fraction representation of b / d. We extend this result to the case where u v ≥ 5. Additionally, we compute [ G u , v : G u , v ] where G u , v = { [ 1 + u v n 1 v n 2 u n 3 1 + u v n 4 ] ∈ SL 2 (Z) : (n 1 , n 2 , n 3 , n 4) ∈ Z 4 } for u , v ≥ 1 , extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for G u , v with u , v ≥ 1. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p -adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m ∈ N such that p ℓ α is an infinite loop mod p m , for all ℓ ∈ N ∪ { 0 }. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional superalgebra with superautomorphism φ over a field of characteristic zero. In [11] the authors gave a positive answer to the Amitsur's conjecture in this setting showing that the φ -exponent of A exists and it is an integer. In the present paper we extend the notion of minimal variety to the context of superalgebras with superautomorphism and prove that a variety is minimal of fixed φ -exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z 2 -grading and superautomorphism. Along the way, we give a contribution on the isomorphism question within the theory of polynomial identities. [ABSTRACT FROM AUTHOR]

For a group G and a positive integer n write B n (G) = { x ∈ G : | x G | ≤ n }. If s ≥ 1 and w is a group word, say that G satisfies the (n , s) -covering condition with respect to the word w if there exists a subset S ⊆ G such that | S | ≤ s and all w -values of G are contained in B n (G) S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n , s) -covering condition with respect to the word w. Then G has a soluble subgroup T such that [ G : T ] and the derived length of T are both (k , n , s) -bounded. (Theorem 1.1.) Let k ≥ 1 and G be a group satisfying the (n , s) -covering condition with respect to the word γ k . Then (1) γ 2 k − 1 (G) has a subgroup T such that [ γ 2 k − 1 (G) : T ] and | T ′ | are both (k , n , s) -bounded; and (2) G has a nilpotent subgroup U such that [ G : U ] and the nilpotency class of U are both (k , n , s) -bounded. (Theorem 1.2.) [ABSTRACT FROM AUTHOR]

It is well known that, over an algebraically closed field k of characteristic zero, for any three integers a , b , c ≥ 2 , any Pham-Brieskorn surface B (a , b , c) : = k [ X , Y , Z ] / (X a + Y b + Z c) is rigid when at most one of a , b , c is 2 and stably rigid when 1 a + 1 b + 1 c ≤ 1. In this paper we consider Pham-Brieskorn domains over an arbitrary field k of characteristic p ≥ 0 and give sufficient conditions on (a , b , c) for which any Pham-Brieskorn domain B (a , b , c) is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on k [ X , Y , Z , T ] / (X m Y + T p r q + Z p e ) = k [ x , y , z , t ] , for m , q > 1 , p ∤ m q and e > r ≥ 1 , fixing y , a crucial result used in [10] to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine 3-space in positive characteristic. We also provide a sufficient condition for B (a , b , c) to be stably rigid. Along the way we prove that for integers a , b , c ≥ 2 with g c d (a , b , c) = 1 and for F (Y) ∈ k [ Y ] , the ring k [ X , Y , Z ] / (X a Y b + Z c + F (Y)) is a rigid domain. [ABSTRACT FROM AUTHOR]

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if char D ≠ 2 , the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) on D ∖ { 0 } with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if char D = 2. [ABSTRACT FROM AUTHOR]

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete "stick breaking") subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior. [ABSTRACT FROM AUTHOR]

Let D , k be integers with D square free and k even. Let N be a positive integer so that (N , D) = 1 when D has residue one modulo four and (N , 4 D) = 1 when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum S X (α , β ; π) attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by D is computed. First a Voronoi summation formula is derived that expresses S X (α , β ; π) in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of S X (α , β ; π) as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N , which is a special case of the multiplicity one theorem. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

Nash-Williams' Strong Immersion Conjecture states that graphs are well-quasi-ordered by the strong immersion relation. That is, given infinitely many graphs, one graph contains another graph as a strong immersion. In this paper we study the analogous problem for directed graphs. It is known that digraphs are not well-quasi-ordered by the strong immersion relation, but for all known such infinite antichains, paths that change direction arbitrarily many times can be found. This paper proves that the converse statement is true: for every positive integer k , the digraphs that do not contain a path that changes direction k times are well-quasi-ordered by the strong immersion relation, even when vertices are labeled by a well-quasi-order. This result is optimal for classes of digraphs closed under taking subgraphs since paths that change direction arbitrarily many times with vertex-labels form an infinite antichain with respect to the strong immersion relation. [ABSTRACT FROM AUTHOR]

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ (R) of R is the cardinality of a minimal cover, and a ring R is called σ -elementary if σ (R) < σ (R / I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σ u (R) to be the size of a minimal cover of R by subrings that contain 1 R (if such a cover exists), and R is σ u -elementary if σ u (R) < σ u (R / I) for every nonzero two-sided ideal of R. In this paper, we classify all σ -elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σ u -elementary rings and prove σ u (R) = σ (R) for every σ u -elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R ′ such that σ (R) = σ u (R ′) , which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if E (N) : = { m : m ⩽ N , σ (R) = m for some ring R } , then we show that | E (N) | = Θ (N / log ⁡ N) , which proves that almost all integers are not covering numbers of a ring. [ABSTRACT FROM AUTHOR]

In a paper from twenty years ago, Brown and Myers proved the clean and uniform result that if m > 1 is an integer, then every curve of the form y 2 = x 3 − x + m 2 has Mordell-Weil rank at least 2. In this article, we further our investigation into the more general family of curves, in which the polynomial x 3 − x is replaced by an arbitrary cubic which is squarefree and has three integral roots. In particular, a new strategy will unveil a subfamily of curves with surprisingly large average rank, and one startling example. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group. Let F (G) denote the monoid of all sequences over G. For Ω ⊂ F (G) , let d Ω (G) be the smallest integer t such that every sequence S over G of length | S | ≥ t has a subsequence in Ω. A sequence S over G is a weak-regular sequence if v g (S) ≤ ord (g) for every g ∈ G. Let N (G) be the smallest integer t such that every weak-regular sequence S over G of length | S | ≥ t has a nontrivial zero-sum subsequence T with v g (T) = v g (S) > 0 for some g | S. The invariant N (G) was formulated in a recent paper [4] to study d Ω (G) and has its own interest, and N (G) has been determined for the cyclic groups of prime order there. In this paper, among other results, we shall determine the precise value of N (G) for cyclic groups of prime power order and for elementary abelian groups of rank two. [ABSTRACT FROM AUTHOR]

Let k , m , n be positive integers with k ⩾ 2. A k -multiset of [ n ] m is a collection of k integers from the set { 1 , 2 , ... , n } in which the integers can appear more than once but at most m times. A family of such k -multisets is called an intersecting family if every pair of k -multisets from the family have non-empty intersection. A finite sequence of real numbers (a 1 , a 2 , ... , a n) is said to be unimodal if there is some k ∈ { 1 , 2 , ... , n } , such that a 1 ⩽ a 2 ⩽ ... ⩽ a k − 1 ⩽ a k ⩾ a k + 1 ⩾ ... ⩾ a n. Given m , n , k , denote C k , ℓ as the coefficient of x k in the generating function (∑ i = 1 m x i) ℓ , where 1 ⩽ ℓ ⩽ n. In this paper, we first show that the sequence of (C k , 1 , C k , 2 , ... , C k , n) is unimodal. Then we use this as a tool to prove that the intersecting family in which every k -multiset contains a fixed element attains the maximum cardinality for n ⩾ k + ⌈ k / m ⌉. In the special case when m = 1 and m = ∞ , our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy [11] , respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem. [ABSTRACT FROM AUTHOR]

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k , the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this paper, we prove that this obvious necessary condition is "nearly" sufficient for graphs with no edge-cut of order 3: for every graph H , every H -immersion free graph with no edge-cut of order 3 can be obtained by an edge-sum of graphs, where each of the summands is obtained from a graph violating the obvious degree condition by adding a bounded number of edges. The condition for having no edge-cut of order 3 is necessary. A simple application of this theorem shows that for every graph H of maximum degree d ≥ 4 , there exists an integer c such that for every positive integer m , there are at most c m unlabeled d -edge-connected H -immersion free m -edge graphs with no isolated vertex, while there are superexponentially many unlabeled (d − 1) -edge-connected H -immersion free m -edge graphs with no isolated vertex. Our structure theorem will be applied in a forthcoming paper about determining the clustered chromatic number of the class of H -immersion free graphs. [ABSTRACT FROM AUTHOR]

Let R = k [ x , y , z ] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals I ⊆ R defining compressed rings with socle k (− s) ℓ ⊕ k (− 2 s + 1) , where s ⩾ 3 and ℓ ⩾ 1 are integers. The case for ℓ = 1 was previously studied in [8] ; a generically minimal resolution was constructed for all such ideals. The paper [7] generalizes this resolution in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals I such that R / I is a ring of Tor algebra class G (r) for some fixed r ⩾ 2 , and R / I may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman in [5]. [ABSTRACT FROM AUTHOR]

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f (x) = x 6 − m ∈ Z [ x ] , it was assumed that the rational integer m ≠ ∓ 1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of Z K ; the ring of integers of K. In particular, we characterize when Z K = Z [ α ] , that is when Z K is monogenic and generated by α. We give sufficient conditions on m , which warranty that K is not monogenic. We finish the paper by investigating the case, where m = e 5 and e ≠ ∓ 1 is a square free rational integer. [ABSTRACT FROM AUTHOR]

For any positive integer t , a t-broom is a graph obtained from K 1 , t + 1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t -brooms, we have χ (G) = o (ω (G) t + 1) , where χ (G) and ω (G) are the chromatic number and clique number of G , respectively. When t = 2 , this answers a question of Schiermeyer and Randerath. Moreover, for t = 2 , we strengthen the bound on χ (G) to 7 ω (G) 2 , confirming a conjecture of Sivaraman. For t ≥ 3 and { t -broom, K t , t }-free graphs, we improve the bound to o (ω t). [ABSTRACT FROM AUTHOR]

This paper elaborates on a sieving technique that has first been applied in 2018 for improving bounds on deterministic integer factorization. We will generalize the sieve in order to obtain a polynomial-time reduction from integer factorization to a specific instance of the multiple choice subset-sum problem. As an application, we will improve upon special purpose factorization algorithms for integers composed of divisors with small difference. In particular, we will refine the runtime complexity of Fermat's factorization algorithm by a large subexponential factor. Our first procedure is deterministic, rigorous, easy to implement and has negligible space complexity. Our second procedure is heuristically faster than the first, but has non-negligible space complexity. [ABSTRACT FROM AUTHOR]

A generalization of the well–known Fibonacci sequence is the ℓ -Fibonacci sequence (F m (ℓ)) m whose first ℓ terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding ℓ terms. The k -Pell sequence (P n (k)) n , which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation P n (k) = F m (ℓ) in positive integers n , k , m , ℓ with k , ℓ ≥ 2. This paper continues and extends prior results which dealt with the above problem for some particular cases of k and ℓ. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence. [ABSTRACT FROM AUTHOR]

By using the work of Frantzikinakis and Wierdl, we can see that for all d ∈ N , α ∈ (d , d + 1) , and integers k ≥ d + 2 and r ≥ 1 , there exist infinitely many n ∈ N such that the sequence (⌊ (n + r j) α ⌋) j = 0 k − 1 is represented as ⌊ (n + r j) α ⌋ = p (j) , j = 0 , 1 , ... , k − 1 , by using some polynomial p (x) ∈ Q [ x ] of degree at most d. In particular, the above sequence is an arithmetic progression when d = 1. In this paper, we show the asymptotic density of such numbers n as above. When d = 1 , the asymptotic density is equal to 1 / (k − 1). Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate asymptotically the sum ∑ d ≤ X (∑ n ≤ Y μ (n) (8 d n)) 2 , where (8 d n) is the Kronecker symbol and d runs over positive, odd, square-free integers. [ABSTRACT FROM AUTHOR]

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j -invariant is ≫ x log − 1 2 ⁡ (x). In this paper, we assume the abc -conjecture to show the sharp asymptotic ∼ c x log − 1 2 ⁡ (x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve E / Q with short Weierstrass equation y 2 = f (x) for reducible f ∈ Z [ x ] , we show that the number of integers d , | d | ≤ D , for which the quadratic twist d y 2 = f (x) has an integral non-2-torsion point is at most D 2 3 + o (1) , assuming the abc -conjecture. (2) We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O (x 1 − ϵ) for some ϵ > 0. We also exhibit a matching lower bound. If instead of the abc -conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias. [ABSTRACT FROM AUTHOR]

We consider a 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces. For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC. In the context of the bilinear Schrödinger equation, the first drift, quantified by the H − 1 -norm of the control, was already observed in [8] and used to deny STLC with controls small in L ∞. In this article, we improve this result by denying STLC with controls small in W − 1 , ∞. Furthermore, for any positive integer n , we formulate assumptions under which one may observe a quadratic drift quantified by the H − n -norm of the control and we use it to deny STLC in suitable spaces. [ABSTRACT FROM AUTHOR]

In this paper, we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers N. We describe the lattice of its ideals for an arbitrary field K and study its derivations over any commutative, unital ring R. [ABSTRACT FROM AUTHOR]

Let e 1 , ... , e c be positive integers and let Y ⊆ P n be the monomial complete intersection defined by the vanishing of x 1 e 1 , ... , x c e c . In this paper, we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points Hilb d (Y) , and discuss applications to the Hilbert scheme of points Hilb d (X) of arbitrary complete intersections X ⊆ P n. [ABSTRACT FROM AUTHOR]

Let 1 ⩽ a < b be two relatively prime integers. Sylvester found that a b − a − b is the largest integer which can not be represented by a x + b y (x , y ∈ Z ⩾ 0) about 160 years ago and this number shall be denoted by g a , b. Let N (a , b) = { n : n ⩽ g a , b , n = a x + b y , x , y ∈ Z ⩾ 0 } and π a , b be the number of primes in N (a , b). Recently, Ramírez Alfonsín and Skałba proved that π a , b ≫ ε g a , b (log ⁡ g a , b) 2 + ε for any fixed ε > 0. They further conjectured that the order of the magnitude of π a , b is 1 2 π (g a , b) , where π (x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a , b with 1 ⩽ a < b and (a , b) = 1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function f : Z + → Z + , such that for every integer g > 0 , every graph of treewidth at least f (g) contains the (g × g) -grid as a minor. For every integer g > 0 , let f (g) be the smallest value for which the theorem holds. Establishing tight bounds on f (g) is an important graph-theoretic question. Robertson and Seymour showed that f (g) = Ω (g 2 log ⁡ g) must hold. For a long time, the best known upper bounds on f (g) were super-exponential in g. The first polynomial upper bound of f (g) = O (g 98 poly log ⁡ g) was proved by Chekuri and Chuzhoy. It was later improved to f (g) = O (g 36 poly log ⁡ g) , and then to f (g) = O (g 19 poly log ⁡ g). In this paper we further improve this bound to f (g) = O (g 9 poly log ⁡ g). We believe that our proof is significantly simpler than the proofs of the previous bounds. Moreover, while there are natural barriers that seem to prevent previous methods from yielding tight bounds for the theorem, it seems conceivable that the techniques proposed in this paper can lead to even tighter bounds on f (g). [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n , k , r , s with n ≥ r k and 2 ≤ s ≤ r , the r -uniform general Kneser hypergraph KG s r (n , k) , has all k -subsets of { 1 , ... , n } as the vertex set and all multi-sets { A 1 , ... , A r } of k -subsets with s -wise empty intersections as the edge set. The case r = s = 2 , was considered by Kneser [7] in 1955, where he conjectured that its chromatic number is n − 2 (k − 1). This was finally proved by Lovász [9] in 1978. The case r > 2 and s = 2 , was considered by Erdös in 1973, and he conjectured that its chromatic number is ⌈ n − r (k − 1) r − 1 ⌉. This conjecture was proved by Alon, Frankl and Lovász [2] in 1986. The case where s > 2 , was considered by Sarkaria [11] in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler [14] in 2006 in the induction method of Sarkaria on the number of prime factors of r , and Sarkaria's proof only worked when s is less than the smallest prime factor of r or s = 2. Finally in 2019, Aslam, Chen, Coldren, Frick and Setiabrata [3] were able to extend this by using methods different from Sarkaria to the case when r = 2 α 0 p 1 α 1 ... p t α t and 2 ≤ s ≤ 2 α 0 (p 1 − 1) α 1 ... (p t − 1) α t . In this paper, by applying the Z p -Tucker lemma of Ziegler [13] and Meunier [10] , we finally prove the general Erdös conjecture and prove the claimed result of Sarkaria for any 2 ≤ s ≤ r. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovász [2] and compute the connectivity of certain simplicial complexes that might be of interest in their own right. [ABSTRACT FROM AUTHOR]

This paper consists of two major parts. In the first part, the relations between Tutte orientations and circular flows are explored. Tutte orientation (modulo orientation) was first observed by Tutte for the study of 3-flow problem, and later extended by Jaeger for circular (2 + 1 / p) -flows. In this paper, it is extended for circular λ -flows for all rational numbers λ. This theorem is one of the key tools in the second part of the paper. It was proved by Lovász et al. (2013) [8] that every (6 p + 1) -odd-edge-connected graph admits a circular (2 + 1 / p) -flow. In the second part, this result is further extended to other odd integers 6 p + 3 and 6 p − 1 for any positive integer p. We show that (i) every (6 p − 1) -odd-edge-connected graph admits a circular (2 + 2 2 p − 1) -flow , and (ii) every (6 p + 3) -odd-edge-connected graph has flow index strictly less than 2 + 1 / p. Both (i) and (ii) generalize some early results. For example, the case p = 1 of (i) is the well known 4-flow theorem of Jaeger (1979) [4] , and the case p = 1 of (ii) is a recent result by Thomassen and the authors (2018) [7]. The proofs of (i) and (ii) provide (principally different) new approach to those previous results. [ABSTRACT FROM AUTHOR]

Let G be a K p -free graph. We say e is a K p -saturating edge of G if e ∉ E (G) and G + e contains a copy of K p. Denote by f p (n , m) the minimum number of K p -saturating edges that an n -vertex K p -free graph with m edges can have. Erdős and Tuza conjectured that f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) n 2 16. Balogh and Liu disproved this by showing f 4 (n , ⌊ n 2 / 4 ⌋ + 1) = (1 + o (1)) 2 n 2 33. They believed that a natural generalization of their construction for K p -free graph should also be optimal and made a conjecture that f p + 1 (n , ex (n , K p) + 1) = (2 (p − 2) 2 p (4 p 2 − 11 p + 8) + o (1)) n 2 for all integers p ≥ 3. The main result of this paper is to confirm the above conjecture of Balogh and Liu. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let anz (G) be the average number of zeros of characters of G , that is the number of zeros in the character table of G divided by the number of irreducible characters of G. In this paper, we classify the finite groups G with anz (G) ≤ 1. In particular, we show that for a rational number a ≤ 1 , there exists a finite group G with anz (G) = a if and only if a ∈ { 0 , 2 3 , 4 5 , 1 } , or a = k k + 2 for some positive integer k. [ABSTRACT FROM AUTHOR]

Given a positive integer r , the r -color size-Ramsey number of a graph H , denoted by R ˆ (H , r) , is the smallest number of edges in a graph G such that in any edge coloring of G with r colors, G contains a monochromatic copy of H. Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle C n is linear in n i.e. R ˆ (C n , r) ≤ f (r) n , for some function f (r). Their proof, however, is based on the Szemerédi's regularity lemma and no explicit function f (r) is given there. Javadi, Khoeini, Omidi and Pokrovskiy gave an alternative proof for this result which avoids using the regularity lemma. Indeed, they proved that f (r) can be taken to be exponential and doubly exponential in r , in case n is even and odd, respectively. In this paper, we improve the upper bound f (r) to a polynomial function in r when n is even and to an exponential function in r when n is odd. We also prove that in the latter case, it cannot be improved to a polynomial bound in r. More precisely, we prove that there are some positive constants c , c ′ such that for every even integer n , we have c r 2 n ≤ R ˆ (C n , r) ≤ c ′ r 120 (log 2 ⁡ r) n and for every odd integer n , we have c 2 r n ≤ R ˆ (C n , r) ≤ c ′ r 2 2 16 r 2 n. [ABSTRACT FROM AUTHOR]

For a number field F and an odd prime number p , let F ˜ be the compositum of all Z p -extensions of F and Λ ˜ the associated Iwasawa algebra. Let G S (F ˜) be the Galois group over F ˜ of the maximal extension which is unramified outside p -adic and infinite places. In this paper we study the Λ ˜ -module X S (− i) (F ˜) : = H 1 (G S (F ˜) , Z p (− i)) and its relationship with X (F ˜ (μ p)) (i − 1) Δ , the Δ : = Gal (F ˜ (μ p) / F ˜) -invariant of the Galois group over F ˜ (μ p) of the maximal abelian unramified pro- p -extension of F ˜ (μ p). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ ˜ -module X (F ˜ (μ p)) (i − 1) Δ is implied by the existence of a Z p d -extension L with X S (− i) (L) : = H 1 (G S (L) , Z p (− i)) being without torsion over the Iwasawa algebra associated to L , and which contains a Z p -extension F ∞ satisfying H 2 (G S (F ∞) , Q p / Z p (i)) = 0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i ≡ 1 mod [ F (μ p) : F ]. This existence is fulfilled for (p , i) -regular fields. [ABSTRACT FROM AUTHOR]

χ -bounded classes are studied here in the context of star colorings and, more generally, χ p -colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ 2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p -boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p -bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most φ (g , ω (G)) | G | holes of length g. [ABSTRACT FROM AUTHOR]

In this paper, we investigate exceptional sets in the biquadratic Waring-Goldbach problem. For example, we prove that all but O (N 60 / 61) integers subject to the necessary local conditions can be represented as the sum of seven biquadrates of primes. Furthermore, the method in this paper can be easily used for exceptional sets for sums of fifth and higher of powers of primes. For a video summary of this paper, please visit https://youtu.be/PGa8P-Px8bk. [ABSTRACT FROM AUTHOR]

For a positive integer k , let f (k) denote the largest integer t such that for every finite abelian group G and every zero-sum free subset S of G , if | S | = k then | Σ (S) | ≥ t. In this paper, we prove that f (k) ≥ 1 6 k 2 , which significantly improves a result of J.E. Olson. We also supply some interesting results on f (k). For a video summary of this paper, please visit https://youtu.be/ZEHZFRUVJKY. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let { F n (k) } n ≥ 2 − k be the k –generalized Fibonacci sequence which starts with 0 , ... , 0 , 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − d y 2 = ± 1 , which is a k –generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k = 2 and [17] for the case k = 3. [ABSTRACT FROM AUTHOR]

For integer k ≥ 2 and prime power q , the algebraic bipartite graph D (k , q) proposed by Lazebnik and Ustimenko in 1993 is useful not only in extremal graph theory but also in coding theory and cryptography. This graph is q -regular, edge-transitive and of girth at least k + 4. Its exact girth g = g (D (k , q)) was conjectured in 1995 to be k + 5 for odd k and q ≥ 4. This conjecture was shown to be valid in 2016 when k + 5 2 | p (q − 1) , where p is the characteristic of F q and m | p n means that m divides p r n for some nonnegative integer r. In this paper, for t ≥ 1 , we prove that (a) g (D (4 t + 2 , q)) = g (D (4 t + 1 , q)) ; (b) g (D (4 t + 3 , q)) = 4 t + 8 if g (D (2 t , q)) = 2 t + 4 ; (c) g (D (8 t , q)) = 8 t + 4 if g (D (4 t − 2 , q)) = 4 t + 2 ; (d) g (D (2 s + 2 t − 5 , q)) = 2 s + 2 t if p ≥ 3 , 2 s | (q − 1) , 2 s + 1 ∤ (q − 1) , 2 ∤ t and t | p (q − 1). A simple upper bound for the girth of D (k , q) is proposed in the end of this paper. [ABSTRACT FROM AUTHOR]

For an arbitrary real number α and a positive integer n , the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined by L n (α) (x) = (− 1) n ∑ j = 0 n (n + α) (n − 1 + α) ⋯ (j + 1 + α) (n − j) ! j ! (− x) j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A 0 ∪ A ∞ of integer pairs (n , α) for which the discriminant of L n (α) (x) is a nonzero square, where A 0 is finite and A ∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α ≠ − 1 , the only pair (n , α) ∈ A ∞ for which the associated Galois group of L n (α) (x) is not A n is (4 , 23). In this paper, we verify this conjecture for (α , n) ∈ A ∞ with α ∈ { − 2 n , − 2 n − 2 , − 2 n − 4 }. In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials L n (α) (x) for n ∈ N and integers α such that α ∈ [ − 2 n − 4 , − 2 n ]. The case α = − 2 n − 1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results. [ABSTRACT FROM AUTHOR]