In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$, generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

Let $k \geqslant 2$ be an integer. We prove that factorisation of integers into k parts follows the Dirichlet distribution $\mathrm{Dir}\left({1}/{k},\ldots,{1}/{k}\right)$ by multidimensional contour integration, thereby generalising the Deshouillers–Dress–Tenenbaum (DDT) arcsine law on divisors where $k=2$. The same holds for factorisation of polynomials or permutations. Dirichlet distribution with arbitrary parameters can be modelled similarly.

We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell $-adic cohomology is Galois invariant for all $\ell $.

Consider the family of automorphic L-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted L-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.

We say that $S\subseteq \mathbb Z$ is a set of k-recurrence if for every measure-preserving transformation T of a probability measure space $(X,\mu )$ and every $A\subseteq X$ with $\mu (A)>0$, there is an $n\in S$ such that $\mu (A\cap T^{-n} A\cap T^{-2n}\cap \cdots \cap T^{-kn}A)>0$. A set of $1$-recurrence is called a set of measurable recurrence. Answering a question of Frantzikinakis, Lesigne, and Wierdl [Sets of k-recurrence but not (k+1)-recurrence. Ann. Inst. Fourier (Grenoble) 56(4) (2006), 839–849], we construct a set of $2$-recurrence S with the property that $\{n^2:n\in S\}$ is not a set of measurable recurrence.

Leonetti and Luca [‘On the iterates of the shifted Euler’s function’, Bull. Aust. Math. Soc., to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$, where $x_1,x_2\geq 1$, $k\geq 0$ and $2 \mid k$, is bounded by $4^{X^{3^{k+1}}}$, where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$, where $X=x_1+x_2+2k$.

The classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power q. For fixed d, we restrict to moduli q so that there is a unique subgroup of invertible classes modulo q of order d. We study distribution properties of these families of sums as q grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions.

Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$, we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $

In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.

We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from algebraic geometry and additive combinatorics.

For integers a and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$. The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

We consider three different families of Vafa–Witten invariants of $K3$ surfaces. In each case, the partition function that encodes the Vafa–Witten invariants is given by combinations of twisted Dedekind η-functions. By utilizing known properties of these η-functions, we obtain exact formulae for each of the invariants and prove that they asymptotically satisfy all higher-order Turán inequalities.

In this note, we prove that every Salem number is expressible as a difference of two Pisot numbers. More precisely, we show that for each Salem number α of degree d, there are infinitely many positive integers n for which $\alpha^{2n-1}-\alpha^n+\alpha$ and $\alpha^{2n-1}-\alpha^n$ are both Pisot numbers of degree d and that the smallest such n is at most $6^{d/2-1}+1$. We also prove that every real positive algebraic number can be expressed as a quotient of two Pisot numbers. Earlier, Salem himself had proved that every Salem number can be written in this way.

For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $a\leqslant a'$ and $a, a'\in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'\in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=\sum_{i\in S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=\emptyset$. For real number x, let $A(-x,x)$ be the number of elements $a\in A$ with $-x\leqslant a\leqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)\geqslant 1$ for all sufficiently large n, then $\limsup_{n\rightarrow\infty}r_A(n)=\infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2\log x/\log 5+c_1\leqslant A(-x,x)\leqslant 2\log x/\log 3+c_2$ for all $x\geqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$ and $A(-x,x) \gt (4/\log 5)\log\log x+c$ for all $x\geqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.

We use the theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for ${\mathrm {GL}}(2)$. We show that the resulting comparison is in fact an expression of an exotic symmetric monoidal structure on the category of ${\mathrm {GL}}(2)$-modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for ${\mathrm {GL}}(2)$, and demonstrate its basic properties.

This paper is a part of the author’s thesis [4].

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.