We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $-distance by polynomial phase functions of degree $k-1$. This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $-approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any finite set of numbers $B.$ The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.

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,

$$\begin{align*}\scriptsize\#\{n\in S\colon n\le M\}\le A\cdot (1+\log M)^{\dim V}.\end{align*}$$

We show that in a parametric family of linear recurrence sequences $a_1(\alpha ) f_1(\alpha )^n + \cdots + a_k(\alpha ) f_k(\alpha )^n$ with the coefficients $a_i$ and characteristic roots $f_i$, $i=1, \ldots ,k$, given by rational functions over some number field, for all but a set of elements $\alpha $ of bounded height in the algebraic closure of ${\mathbb Q}$, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.

The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A.

Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define Sd={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |Sd| is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |Sd|>(α3-ϵ)N2?

We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets A with |Sd|>Cα3.13N2 for all d ≠ 0, where C is an absolute constant. We also show that in the special case where $G = {\mathbb{F}}_2^n$, one can always find a d with |Sd|>(α4-ϵ)N2.

A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.

We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form

$$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$, and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$, see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

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.

Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral.

Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have

$$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$

where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.

The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$. This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1 < c < 2$, is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.

In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $. We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

We examine a recursive sequence in which $s_n$ is a literal description of what the binary expansion of the previous term $s_{n-1}$ is not. By adapting a technique of Conway, we determine the limiting behaviour of $\{s_n\}$ and dynamics of a related self-map of $2^{\mathbb {N}}$. Our main result is the existence and uniqueness of a pair of binary sequences, each the complement-description of the other. We also take every opportunity to make puns.

For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G.

Recently, Mandache proved a “popular differences” version of this result in the finite field case $G = {\mathbb{F}}_p^n$, showing that for any subset A ⊂ G × G of density α, one can fix d ≠ 0 such that A contains a large fraction, now known to be approximately α4, of all corners with difference d, as x, y vary over G. We generalise Mandache’s result to all compact abelian groups G, as well as the case of corners in $\mathbb{Z}^2$.

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline {n}\in \mathbb {Z}_m$, let $R_S(\overline {n})$ denote the number of solutions of the equation $\overline {n}=\overline {s}+\overline {s'}$ with ordered pairs $(\overline {s},\overline {s'})\in S^2$. We determine the structure of $A,B\subseteq \mathbb {Z}_m$ with $|(A\cup B)\setminus (A\cap B)|=m-2$ such that $R_{A}(\overline {n})=R_{B}(\overline {n})$ for all $\overline {n}\in \mathbb {Z}_m$, where m is an even integer.

In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:

$$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$