We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. We consider the skew products on $\mathbb {T} \times {\Gamma \backslash G}$ and prove that the Möbius function is linearly disjoint from these skew products which improves the recent result of Huang, Liu and Wang [‘Möbius disjointness for skew products on a circle and a nilmanifold’, Discrete Contin. Dyn. Syst. 41(8) (2021), 3531–3553].

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of $\mathbb {R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb {R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order $p^{m}q^{n}$ , when one of the exponents is $\leq 6$ or when $p^{m-2}<q^{4}$ , and also prove that a tiling subset of a cyclic group of order $p_{1}^{m}p_{2}\dotsm p_{n}$ is spectral.

Let $[t]$ be the integral part of the real number t. We study the distribution of the elements of the set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ in the arithmetical progression $\{a+dq\}_{d\geqslant 0}$ . We give an asymptotic formula

$$ \begin{align*} S(x; q, a) := \sum_{\substack{m\in \mathcal{S}(x)\\ m\equiv a \pmod q}} 1 = \frac{2\sqrt{x}}{q} + O((x/q)^{1/3}\log x), \end{align*} $$

which holds uniformly for $x\geqslant 3$ , $1\leqslant q\leqslant x^{1/4}/(\log x)^{3/2}$ and $1\leqslant a\leqslant q$ , where the implied constant is absolute. The special case $S(x; q, q)$ confirms a recent numerical test of Heyman [‘Cardinality of a floor function set’, Integers 19 (2019), Article no. A67].

Let $f(x)\in \mathbb {Z}[x]$ be a nonconstant polynomial. Let $n\ge 1, k\ge 2$ and c be integers. An integer a is called an f-exunit in the ring $\mathbb {Z}_n$ of residue classes modulo n if $\gcd (f(a),n)=1$ . We use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,\ldots ,x_k)$ of the congruence $x_1+\cdots +x_k\equiv c\pmod n$ with all $x_i$ being f-exunits in the ring $\mathbb {Z}_n$ . This extends a recent result of Anand et al. [‘On a question of f-exunits in $\mathbb {Z}/{n\mathbb {Z}}$ ’, Arch. Math. (Basel)116 (2021), 403–409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$-periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

En s’appuyant sur la notion d’équivalence au sens de Bohr entre polynômes de Dirichlet et sur le fait que sur un corps quadratique la fonction zeta de Dedekind peut s’écrire comme produit de la fonction zeta de Riemann et d’une fonction L, nous montrons que, pour certaines valeurs du discriminant du corps quadratique, les sommes partielles de la fonction zeta de Dedekind ont leurs zéros dans des bandes verticales du plan complexe appelées bandes critiques et que les parties réelles de leurs zéros y sont denses.

Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum

$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$

$(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $

$$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$

$\hat t:{\mathbb F_p} \to \mathbb C$

$\varepsilon > 0$

$a \in \mathbb F_p^ \times $

$${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$

$$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$

$p \to \infty $

${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$

We prove a lower bound for the large sieve with square moduli.

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences, such as the sum of digits in base $q$ modulo $m$, the Rudin–Shapiro sequence, and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$-function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$-adically analytic phases, which can be naturally seen as a $p$-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

We give a purity result for two kinds of exponential sums of the type $\sum_{x\in k^n}\psi(f(x))$, where k is a finite field of characteristic p and $\psi:k\to{\mathbb C}^\star$ is a non-trivial additive character. In the first case, $f\in k[x_1,\dots,x_n]$ is a polynomial of degree divisible by p whose highest-degree homogeneous form defines a non-singular projective hypersurface, and in the second case, f is a polynomial of degree prime to p whose highest-degree homogeneous form defines a projective hypersurface with isolated singularities.

This paper shows that a nontrivial uniform decay estimate for complete exponential sums modulo pr, determined by a polynomial map ${\bf P} = (P_1, P_2)$ follows from the existence of a ‘good P decomposition’ of ${\mathbb Z}_p^n$, a property that can be proved with geometric methods, and which was introduced in an earlier article by the present author.

Estimates are given for the exponential sum $\sum_{x=1}^p \exp(2\pi i f(x)/p)$, $p$ a prime and $f$ a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod $p$, namely the smallest $\gamma$ such that $f(x_1)+\cdots +f(x_{\gamma})\equiv N$ (mod $p$) is solvable in integers for any $N$.