Search

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model M of PA (Peano arithmetic) is recursively saturated iff M has an expansion that satisfies the subsystem Δ11−CA0 of second order arithmetic. In this paper we identify a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, and we offer a correct proof of the problematic direction. [ABSTRACT FROM AUTHOR]

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p is the same in both fields for almost all prime p. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n we exhibit a positive constant c_{n} such that any two degree n number fields K and L are arithmetically equivalent if and only if the set of primes p such that the arithmetic type of p in K and L is not the same has Dirichlet density at most c_n. We in fact show that \displaystyle c_n=\frac {1}{4n^2} works and give a heuristic evidence that points to the fact that this value might be improved to \displaystyle \frac {2}{n^2}. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

Let K be a complete discrete valuation field with residue class field k, where both are of positive characteristic p. Then the group of wild automorphisms of K can be identified with the group under composition of formal power series over k with no constant term and X-coefficient 1. Under the hypothesis that p > b2, we compute the first nontrivial coefficient of the pth iterate of a power series over k of the form ƒ = X + ∑i ≥ 1 aiXb+i. As a result, we obtain a necessary and sufficient condition for an automorphism to be '' b-ramified'', having lower ramification numbers of the form in(ƒ) = b(1 + ⋅⋅⋅ + pn). This is a vast generalization of Nordqvist's 2017 theorem on 2-ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in Cp and a 2015 result of Lindahl-Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl-Nordqvist's 2018 theorem bounding the norm of periodic points of 2-ramified power series. [ABSTRACT FROM AUTHOR]

In this paper, we use Rouché's theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results and bounds regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the "exclusion region" around a given zero or pole of a rational function in which no critical point may lie. [ABSTRACT FROM AUTHOR]

This article provides a relation between the classical Bohr radius, the arithmetic Bohr radius, and the multidimensional Bohr radius. Through this relation, we give answers to some questions, one is raised by Defant et al. and other one is an open problem, both are related to the multidimensional Bohr radius. Moreover, we provide better new estimates of Bohr radii for holomorphic functions defined on Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we improve Ru's defect relation and the height inequality in the case when X is a normal projective surface and Dj, 1 = j = q, are big and asymptotic free divisors without irreducible common components on X. As a consequence, we derive a sharp result in the qualitative statement. [ABSTRACT FROM AUTHOR]

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well-known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of non-negative real numbers. [ABSTRACT FROM AUTHOR]

Chatterjee and Garg [Proc. Amer. Math. Soc. 151 (2023), pp. 2011-2022] established closed form for a q-analogue of the Euler-Stieltjes constants. In this article, we aim to build upon their work by extending it to a q-analogue of the double zeta function. Specifically, we derive a closed form expression for \gamma _{0,0}(q) which is a q-analogue of Euler's constant of height 2 and appear as the constant term in the Laurent series expansion of a q-analogue of the double zeta function around s_1 = 1 and s_2=1. Moreover, we examine the linear independence of a set of numbers involving the constant \gamma _0^{\prime *}(q^i), where 1 \leq i \leq r for any integer r \geq 1, that appears in the Laurent series expansion of a q-double zeta function. Finally, we discuss the irrationality of certain numbers involving a 2-double Euler-Stieltjes constant (\gamma _{0,0}(2)). [ABSTRACT FROM AUTHOR]

This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions; then we further extend these to hidden symmetries of some hyperbolic link complements. [ABSTRACT FROM AUTHOR]

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups). [ABSTRACT FROM AUTHOR]

We study the arithmetic of division fields of semistable abelian varieties A/Q. The Galois group of Q(A[2])/Q is analyzed when the conductor of A is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in our paper Paramodular abelian varieties of odd conductor. [ABSTRACT FROM AUTHOR]

Harmonic numbers Hk = ∑0 3 be a prime. We show that p-1 ∑ k=1 Hk/k2k = 0 (mod p), p-1 ∑ k=1 H2/k = 2p - 2 (mod p2), p-1 ∑ k=1 H3/k = 6 (mod p), and p-1∑ k=1 H2 k/k2 = 0 (mod p) provided p > 5. (In contrast, it is known that ∑∞k=1 Hk/(k2k) = π2/12 and ∑∞k=1 H2 k/k2 = 17p4/360.) Our tools include some sophisticated combinatorial identities and properties of Bernoulli numbers. [ABSTRACT FROM AUTHOR]

We show that in two dimensional flat torus the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian. [ABSTRACT FROM AUTHOR]

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space \mathbb {H}^n, n\geq 2, we show that: one can produce infinitely many maximal quasi–arithmetic reflection groups acting on \mathbb {H}^2; they admit infinitely many different fields of definition; the degrees of their fields of definition are unbounded. However, for n\geq 14 an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi–arithmetic case. [ABSTRACT FROM AUTHOR]

We present a previously unpublished proof of the conservativity of WKL0 over IΣ1 using the Arithmetized Completeness Theorem, which, in particular, constitutes an ω-interpretation of WKL0 in IΣ1. We also show that WKL0* is interpretable in IΔ0 + exp. [ABSTRACT FROM AUTHOR]

The systole of a Riemannian manifold is the minimal length of a non-contractible closed geodesic loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of semisimple x\in \operatorname {SL}_n(\mathbf {R}) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior. [ABSTRACT FROM AUTHOR]

For relatively prime positive integers $u_0$ and $r$, we consider the arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. par Define $L_n:=operatorname {lcm}{u_0, u_1, ..., u_n}$ and let $age 2$ be any integer. In this paper, we show that for integers $alpha , rgeq a$ and $ngeq 2alpha r$, we have $$L_ngeq u_0r^{alpha +a-2}(r+1)^n.$$ In particular, letting $a=2$ yields an improvement to the best previous lower bound on $L_n$ (obtained by Hong and Yang) for all but three choices of $alpha , rgeq 2$. [ABSTRACT FROM AUTHOR]

In this paper we observe that in the hypergraph removal lemma, the edge removal can be done in such a way that the symmetries of the original hypergraph remain preserved. As an application we prove the following generalization of Szemerédi's Theorem on arithmetic progressions. Let $A$ be an Abelian group with subsets $S_1,S_2,dots ,S_t$ such that the number of arithmetic progressions $x,x+d,dots ,x+(t-1)d$ with $x+(i-1)din S_i$ is $o(|A|^2)$. Then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have any arithmetic progression of the above type. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the length of a Lyubeznik resolution of a monomial ideal gives an upper bound for the arithmetical rank of the ideal. [ABSTRACT FROM AUTHOR]

In this paper we consider a non-commutative analogue of the arithmetic-geometric mean inequality $$a^{r}b^{1-r}+(r-1)bgeq ra$$ for two positive numbers $a,b$ and for $r> 1$. We show that under certain assumptions the non-commutative analogue of $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to the $r$-mean. The case $0

In a paper of James Hlavka it is stated that $F(3)+F(2)+F(2)\neq\R$. In this manuscript we show that this is false by establishing that $F(3)\pm F(2)\pm F(2)=\R$. We also describe the corresponding products and quotients. [ABSTRACT FROM AUTHOR]

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Rüssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition. [ABSTRACT FROM AUTHOR]

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-cocompact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically. [ABSTRACT FROM AUTHOR]

We apply the arithmetic theory of Teichmüller modular forms to calculating constants in relations, which are connected with Klein's (amazing) formulas, between certain invariants of canonical curves of genus g = 3,4. [ABSTRACT FROM AUTHOR]

Let S ⊆ N be finite. Is there a positive definite quadratic form that fails to represent only those elements in S? For S = Ø, this was solved (for classically integral forms) by the 15-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the 290-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent S = {m}; of his 88 candidates, he could provide complete justifications for all but one. In the same spirit, we ask, "For which S = {m, n} does there exist a quadratic form excepting only the elements of S?" Extending the techniques of Bhargava and Hanke, we answer this question for quaternary forms. In the process, we provide a new proof of the original outstanding conjecture of Halmos, namely, that x²+2y²+7z²+13w² represents all positive integers except 5. We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the 73 possible pairs that a classically integral positive definite quadratic form may except. [ABSTRACT FROM AUTHOR]

We prove that for every computable limit ordinal α there exists a computable structure A which is Δα0-categorical and α is smallest such, but nonetheless for every isomorphic computable copy B of A there exists a β < α such that A ≅=Δ0βB. This answers a question raised by Kalimullin in personal communication with the third author. [ABSTRACT FROM AUTHOR]

We prove the existence of Riesz bases of exponentials of L2(Ω), provided that Ω � Rd is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case. [ABSTRACT FROM AUTHOR]

In this work we prove that the set of the difference of primes is a Δr*-set. The work is based on the recent dramatic new developments in the study of bounded gaps between primes, reached by Zhang, Maynard and Tao. [ABSTRACT FROM AUTHOR]

We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure laws. We then see that the relatively recently-defined q-bracket operator 〈f〉 q, studied by Bloch-Okounkov, Zagier, and others for its quasimodular properties, plays a deep role in the theory of partitions, quite apart from questions of modularity. Moreover, we give an explicit formula for the coefficients of 〈f〉 q for any function f defined on partitions, and, conversely, give a partition-theoretic function whose q-bracket is a given power series. [ABSTRACT FROM AUTHOR]

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices. [ABSTRACT FROM AUTHOR]

We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p-harmonic type in Carnot groups of Heisenberg type for 1 < p < ∞. In the presentation we exhibit the special cases of sub-Laplace equations (p = 2) and the case p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems. [ABSTRACT FROM AUTHOR]

Let (M, g, e-f dv) be a complete smooth metric measure space with Bakry-Émery Ricci curvature nonnegative outside a compact set. We prove that the number of ends of such a measure space is finite if f has at most sublinear growth outside the compact set. In particular, we give an explicit upper bound for the number. [ABSTRACT FROM AUTHOR]

We show that Friedman's finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris--Harrington theorem in PA. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension d is equivalent to the Paris--Harrington theorem for dimension d + 1 . [ABSTRACT FROM AUTHOR]

We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor-space exists. As a base system we take ACA0ω +(μ). The system ACA0ω is the higher order extension of Friedman's system ACA0, and (μ) denotes Feferman's μ, that is, a uniform functional for arithmetical comprehension defined by f(μ(f)) = 0 if ∃nf(n) = 0 for f ∈ ℕℕ. Feferman's μ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reason ACA0ω + (μ) is the weakest fragment of higher order arithmetic where σ-additive measures are directly definable. We obtain that over ACA0ω+ (μ) the existence of the Lebesgue measure is Π2¹ -conservative over ACA0ω and with this conservative over PA. Moreover, we establish a corresponding program extraction result. [ABSTRACT FROM AUTHOR]

Let K be a number field, and let B be a smooth projective curve over K of genus ≤ 1 such that B(K) is infinite. Let A be an Abelian variety defined over the function field K(B). Suppose there are infinitely many nontrivial, pairwise disjoint extensions L/K of bounded degree such that B(L) is infinite and that AP (L)tor ≠= 0 for every point P ∈ B(L) at which the specialization AP is smooth. We show that A(K(B))tor ≠ = 0. If A/K(B) is not a constant Abelian variety over K, the extensions L/K need not have bounded degree, and we can replace B(K) being infinite by B(K) ≠ ∅. This provides evidence in support of a question of Graber-Harris-Mazur-Starr on rational pseudo-sections of arithmetic surjective morphisms. [ABSTRACT FROM AUTHOR]

A closed connected hyperbolic n-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic (n +1)-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension n = 3 using right-angled dodecahedra and 120-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every k ≥ 1, we build an orientable compact closed 3-manifold tessellated by 16k right-angled dodecahedra that bounds a 4-manifold tessellated by 32k right-angled 120-cells. A notable feature of this family is that the ratio between the volumes of the 4-manifolds and their boundary components is constant and, in particular, bounded. [ABSTRACT FROM AUTHOR]

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is S30 - complete. However, when we consider groups of infinite rank, it becomes S1¹ - complete, so it cannot be characterized by a first-order formula in the language of arithmetic. [ABSTRACT FROM AUTHOR]

There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over R or C, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective plane - even one with weak axioms. [ABSTRACT FROM AUTHOR]

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L* is bounded from below outside the set of roots of unity by an absolute constant. We also discuss some features of Property (B): stability by algebraic extensions and relations with field arithmetic. As a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free. [ABSTRACT FROM AUTHOR]

We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms. [ABSTRACT FROM AUTHOR]

We prove that the binomial arithmetical rank of the edge ideal of a forest coincides with its big height. [ABSTRACT FROM AUTHOR]

Central sets in ℕ were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set contains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of homogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of ℕ means that gaps are bounded. Given Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. Kronecker's Theorem says that if 1, α1, α2, ⋯, αv are linearly independent over ℚ and U is a nonempty open subset of (-½ , ½ )v, then {x ε ℕ : (w(α1x), ⋯, w(αvx)) ε U} is nonempty, and Weyl showed that this set has positive density. We show here that if 0̄ is in the closure of U, then this set is strongly central. More generally, let P1, P2, ⋯, Pv be real polynomials with zero constant term. We show that {x ε ℕ : (w(P1(x)), ⋯, w(Pv(x))) ε U} is nonempty for every open U with 0̄ ε …U if and only if it is very strongly central for every such U and we show that these conclusions hold if and only if any nontrivial rational linear combination of P1, P2, ⋯, Pv has at least one irrational coefficient. [ABSTRACT FROM AUTHOR]

This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem. For a given number-theoretic function g, let Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. be the least natural number R such that for all colourings P of the d-element subsets of {0, . . . ,R-1} with at most c colours there exists a subset H of {0, . . . ,R-1} such that P has constant value on all d-element subsets of H and such that the cardinality of H is not smaller than max{k, g(min(H))}. If g is a constant function with value e, then Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is equal to the usual Ramsey number Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and if g is the identity function, then Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is the corresponding Paris-Harrington number, which typically is much larger than Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. In this article we give for all d ≥ 2 a sharp classification of the functions g for which the function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most (d - 1)-quantifiers. Such a quick growth will in particular happen for any function g growing at least as fast as Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (where ε > 0 is fixed) but not for the function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (Here log* denotes the functional inverse of the tower function.) To obtain such results and even sharper bounds we employ certain suitable transfinite iterations of nonconstructive lower bound functions for Ramsey numbers. Thereby we improve certain results from the article A classification of rapidly growing Ramsey numbers (PAMS 132 (2004), 553-561) of the first author, which were obtained by employing constructive ordinal partitions. [ABSTRACT FROM AUTHOR]