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For an integer $b\geq 2$ , a positive integer is called a b -Niven number if it is a multiple of the sum of the digits in its base- b representation. In this article, we show that every arithmetic progression contains infinitely many b -Niven numbers. [ABSTRACT FROM AUTHOR]

The article informs about new effective results in the theory of the Riemann zeta-function, focusing on four groups providing estimates for the zeta-function and associated functions under the assumption of the Riemann hypothesis. Topic include explicit and RH estimates for various functions related to the zeta-function, including their applications to the distribution of prime numbers and other arithmetic properties, emphasizing the importance of these findings in mathematical research.

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$ -extensions of F known as constant $\mathbb {Z}_p$ -extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$ , where K is any field of characteristic $\ell $ , showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$ , where $m,n, d\geq 2$ are integers. [ABSTRACT FROM AUTHOR]

Let $p_t(a,b;n)$ denote the number of partitions of n such that the number of t -hooks is congruent to $a \bmod {b}$. For $t\in \{2, 3\}$ , arithmetic progressions $r_1 \bmod {m_1}$ and $r_2 \bmod {m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males and Ono ['Distributions on partitions arising from Hilbert schemes and hook lengths', Forum Math. Sigma 10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t -cores and t -quotients. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by. We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets. [ABSTRACT FROM AUTHOR]

Andrews [ Generalized Frobenius Partitions , Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers $\ell ,$ all primes $p\geq 5$ and all values $r, 0 such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' memoir, classical q -series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question. [ABSTRACT FROM AUTHOR]

We show that there is a red-blue colouring of [N] with no blue 3-term arithmetic progression and no red arithmetic progression of length eC(log N)3/4 (log log N)1/4. Consequently, the two-colour van der Waerden number w(3,k) is bounded below by kb(k), where b(k) = c(log k/log log k)1/3. Previously it had been speculated, supported by data, that w(3,k) = O(k²). [ABSTRACT FROM AUTHOR]

We prove a quantitative partial result in support of the dynamical Mordell–Lang conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field K of characteristic p, a semiabelian variety X defined over a finite subfield of K and endowed with a regular self-map $\Phi :X{\longrightarrow } X$ defined over K, a point $\alpha \in X(K)$ and a subvariety $V\subseteq X$ , then the set of all nonnegative integers n such that $\Phi ^n(\alpha)\in V(K)$ is a union of finitely many arithmetic progressions along with a subset S with the property that there exists a positive real number A (depending only on X, $\Phi $ , $\alpha $ and V) such that for each positive integer M, $$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$ [ABSTRACT FROM AUTHOR]

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature. [ABSTRACT FROM AUTHOR]

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński. We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered. [ABSTRACT FROM AUTHOR]

The article focus on explicit version of Chen's theorem to improve many mathematical tools that are needed for the task. Topics include making Vinogradov's proof of Goldbach's weak conjecture completely explicit for better approximation of Goldbach's conjecture; and salient steps and results employed to obtain Theorem based on the linear sieve.

Let $\varepsilon> 0$ be sufficiently small and let $0. We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$. This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods. [ABSTRACT FROM AUTHOR]

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula. [ABSTRACT FROM AUTHOR]

The conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth's theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size Θ(√ k) which span k edges. This is best possible and goes far beyond the conjecture. [ABSTRACT FROM AUTHOR]

We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients. [ABSTRACT FROM AUTHOR]

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to $X$ is $\gg X^{1-R}$ , where $R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance's conjectured density of $X^{1-R}$ with $R=(1+o(1))\log \log \log X/\text{log}\log X$. [ABSTRACT FROM AUTHOR]

For each t 2 R, we define the entire function ... where Φ is the super-exponentially decaying function ... Newman showed that there exists a finite constant Λ (the de Bruijn-Newman constant) such that the zeros of Ht are all real precisely when t > Λ. The Riemann hypothesis is equivalent to the assertion Λ 6 0, and Newman conjectured the complementary bound Λ > 0. In this paper, we establish Newman's conjecture. The argument proceeds by assuming for contradiction that Λ < 0 and then analyzing the dynamics of zeros of Ht (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of Ht in the range Λ < t 6 0, until one establishes that the zeros of H0 are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery. [ABSTRACT FROM AUTHOR]

An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1. [ABSTRACT FROM AUTHOR]

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$ -near primitive root modulo $p$. We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$ -near primitive root and we prove a more difficult variant. [ABSTRACT FROM AUTHOR]

Motivated by Ramanujan's continued fraction and the work of Richmond and Szekeres ['The Taylor coefficients of certain infinite products', Acta Sci. Math. (Szeged) 40 (3–4) (1978), 347–369], we investigate vanishing coefficients along arithmetic progressions in four quotients of infinite product expansions and obtain similar results. For example, $a_{1}(5n+4)=0$ , where $a_{1}(n)$ is defined by $$\begin{eqnarray}\displaystyle {\displaystyle \frac{(q,q^{4};q^{5})_{\infty }^{3}}{(q^{2},q^{3};q^{5})_{\infty }^{2}}}=\mathop{\sum }_{n=0}^{\infty }a_{1}(n)q^{n}. & & \displaystyle \nonumber\end{eqnarray}$$ [ABSTRACT FROM AUTHOR]

Putnam ['On the non-periodicity of the zeros of the Riemann zeta-function', Amer. J. Math. 76 (1954), 97–99] proved that the sequence of consecutive positive zeros of $\unicode[STIX]{x1D701}(\frac{1}{2}+it)$ does not contain any infinite arithmetic progression. We extend this result to a certain class of zeta functions. [ABSTRACT FROM AUTHOR]

Answering a question of Gowers, Tao proved that any A × B × C ⊂ SL d (𝔽 q )3 contains | A || B || C |/|SL d (𝔽 q )| + Od (|SL d (𝔽 q )|2/ q min(d −1,2)/8) three-term progressions (x , xy , xy 2). Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for PSL2(𝔽 q ) with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when d > 2, but with the error term O (|SL d (𝔽 q )|2/ q (d −1)/24). [ABSTRACT FROM AUTHOR]

Magyar has shown that if B ⊂ ℤ d has positive upper density (d ⩾ 5), then the set of squared distances {|| b 1 − b 2||2 : b 1, b 2 ∈ B } contains an infinitely long arithmetic progression, whose period depends only on the upper density of B. We extend this result by showing that B contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of B and the number of vertices of the sought tree). In particular, B contains all chains of elements with gaps in some given arithmetic progression (which depends on the length of the sought chain). This is a discrete analogue of a result obtained recently by Bennett, Iosevich and Taylor on chains with prescribed gaps in sets of large Haussdorf dimension. Our techniques are Ergodic theoretic and may be of independent interest to Ergodic theorists. In particular, we obtain Ergodic theoretic analogues of recent optimal spherical distribution results of Lyall and Magyar which, via Furstenberg's correspondence principle, recover their combinatorial results. [ABSTRACT FROM AUTHOR]

Magyar has shown that if B ⊂ ℤ d has positive upper density (d ⩾ 5), then the set of squared distances {|| b 1 − b 2||2 : b 1, b 2 ∈ B } contains an infinitely long arithmetic progression, whose period depends only on the upper density of B. We extend this result by showing that B contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of B and the number of vertices of the sought tree). In particular, B contains all chains of elements with gaps in some given arithmetic progression (which depends on the length of the sought chain). This is a discrete analogue of a result obtained recently by Bennett, Iosevich and Taylor on chains with prescribed gaps in sets of large Haussdorf dimension. Our techniques are Ergodic theoretic and may be of independent interest to Ergodic theorists. In particular, we obtain Ergodic theoretic analogues of recent optimal spherical distribution results of Lyall and Magyar which, via Furstenberg's correspondence principle, recover their combinatorial results. [ABSTRACT FROM AUTHOR]

Let G be a finite D-quasirandom group and A ⊂ Gk a δ-dense subset. Then the density of the set of side lengths g of corners $$ \{(a_{1},\dotsc,a_{k}),(ga_{1},a_{2},\dotsc,a_{k}),\dotsc,(ga_{1},\dotsc,ga_{k})\} \subset A $$ converges to 1 as D → ∞. [ABSTRACT FROM AUTHOR]

A collection of k sets is said to form a k-sunflower, or Δ-system, if the intersection of any two sets from the collection is the same, and we call a family of sets F sunflower-free if it contains no 3-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt ('On large subsets of Fn q with no three-term arithmetic progression', Ann. of Math. (2) 185 (2017), 339-343); ('Progression-free sets in Zn4 are exponentially small', Ann. of Math. (2) 185 (2017), 331-337) we apply the polynomial method directly to Erdős-Szemerédi sunflower problem (Erdős and Szemerédi, 'Combinatorial properties of systems of sets', J. Combin. Theory Ser. A 24 (1978), 308-313) and prove that any sunflower-free family F of subsets of {1, 2, . . . , n} has size at most ∣F∣ ≤ 3n X k≤n/3Σ (n k) ≤ ( 3 22/3 )n(1Co.1)) . We say that a set A ⊂ (Z/DZ)n D {1, 2, . . . , D}n for D > 2 is sunflower-free if for every distinct triple x, y, z 2 A there exists a coordinate i where exactly two of xi , yi , zi are equal. Using a version of the polynomial method with characters X : Z/DZ → C instead of polynomials, we show that any sunflower-free set A ⊂ (Z/DZ)n has size ∣A∣ ≤ cnD where cD = 3/22/3 (D - 1)2/3. This can be seen as making further progress on a possible approach to proving the Erdős and Rado sunflower conjecture ('Intersection theorems for systems of sets', J. Lond. Math. Soc. (2) 35 (1960), 85-90), which by the work of Alon et al. ('On sunflowers and matrix multiplication', Comput. Complexity 22 (2013), 219-243; Theorem 2.6) is equivalent to proving that cD ≤ C for some constant C independent of D. [ABSTRACT FROM AUTHOR]

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment. [ABSTRACT FROM AUTHOR]

We generalise the classical Bombieri-Vinogradov theorem for short intervals to a nonabelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extension L/Q exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modular L-function L(s, f), the fundamental discriminants d for which the d-quadratic twist of L(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals. [ABSTRACT FROM AUTHOR]

We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log x)^{{\it\epsilon}}$ containing $\gg _{{\it\epsilon}}\log \log x$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$. [ABSTRACT FROM PUBLISHER]

We show that if A ⊆ {1,...,N} does not contain any nontrivial solutions to the equation x + y + z = 3w, then ∣ A ⩽ N/exp(c(log N)1/7) , where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A + A + A contain long arithmetic progressions if A ⊆ {1,...,N}, or high-dimensional affine subspaces if A ⊆ 픽qn, if A has density of the shape above. [ABSTRACT FROM AUTHOR]

Combining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$. [ABSTRACT FROM PUBLISHER]

We study low level nondefinability in the Turing degrees. We prove a variety of results, including, for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤, REA) where REA stands for the "r.e. in and above" predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1-theory of (D, ≤ REA) is decidable. [ABSTRACT FROM AUTHOR]

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions. [ABSTRACT FROM AUTHOR]

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation} Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation} [ABSTRACT FROM AUTHOR]

We prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$). [ABSTRACT FROM AUTHOR]

Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 픽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average. [ABSTRACT FROM AUTHOR]

We establish Manin's conjecture for a cubic surface split over ℚ and whose singularity type is 2A2 + A1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath–Brown) and draws on the work of Deligne. [ABSTRACT FROM PUBLISHER]

We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput.29 (1975), 79–81] and Sun [‘On integers not of the form ±pa±qb’, Proc. Amer. Math. Soc.128 (2000), 997–1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results. [ABSTRACT FROM PUBLISHER]

In this paper, we study certain pairings which are defined as follows: if A and B are finite subsets of an arbitrary group, a Wakeford–Fan–Losonczy pairing from B onto A is a bijection φ : B → A such that bφ(b) ∉ A, for every b ∈ B. The number of such pairings is denoted by μ(B, A).We investigate the quantity μ(B, A) for A and B, two finite subsets of an arbitrary group satisfying 1 ∉ B, |A| = |B|, and the fact that the order of every element of B is ≥ |B| + 1. Extending earlier results, we show that in this case, μ(B, A) is never equal to 0. Furthermore we prove an explicit lower bound on μ(B, A) in terms of |B| and the cardinality of the group generated by B, which is valid unless A and B have a special form explicitly described. In the case A = B, our bound holds unless B is a translate of a progression. [ABSTRACT FROM PUBLISHER]

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or −1 and if ⊂ ℤ/pℤ is not too large (with respect to p), then | + t ⋅ | is significantly larger than 2|| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | + t ⋅ | ≥ 2.08 ||−2. [ABSTRACT FROM PUBLISHER]

We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fpn. For every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cpn/2 and for almost all sets R of size r of Fpn. Let A be any subset of R of size at least αr; then A contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, Łuczak and Rödl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right. [ABSTRACT FROM PUBLISHER]

Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n by$$ g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)\cdots(b+(n+k)a)}{\operatorname{lcm}(b+na,b+(n+1)a,\dots,b+(n+k)a)}. $$If we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions. [ABSTRACT FROM PUBLISHER]

Let f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 풮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 풮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford. [ABSTRACT FROM AUTHOR]

Assuming a conjecture intermediate in strength between one of Chowla and one of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m ⩾ 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: there exists an infinite arithmetic progression of positive odd numbers such that for any term k of the sequence and any nonnegative integer n, each of the 16 integers k - 2n, k - 2 - 2n, k - 4 - 2n, k - 6 - 2n, k - 8 - 2n, k - 10 - 2n, k - 12 - 2n, k - 14 - 2n, k2n - 1, (k - 2)2n - 1, (k - 4)2n - 1, (k - 6)2n - 1, (k - 8)2n - 1, (k - 10)2n - 1, (k - 12)2n - 1 and (k - 14)2n - 1 has at least two distinct odd prime factors; in particular, for each term k, none of the eight integers k, k - 2, k - 4, k - 6, k - 8, k - 10, k - 12 or k - 14 can be expressed as a sum of two prime powers. [ABSTRACT FROM AUTHOR]

AbstractSuppose that Gis an abelian group and that A? Gis finite and contains no non-trivial three-term arithmetic progressions. We show that |A+A| ??|A|(log|A|)???. [ABSTRACT FROM AUTHOR]