Let s be a fixed positive integer constant and let $\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set ${\mathcal M}\subseteq \mathbb {F}_p^*$ of size $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer $H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$

When $s=1$, we show that, for almost all primes p, if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.

We investigate the sums $(1/\sqrt {H}) \sum _{X < n \leq X+H} \chi (n)$, where $\chi $ is a fixed non-principal Dirichlet character modulo a prime q, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty $ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty $, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $\chi $, even when $H = q/\log ^{A}q$ for any fixed $A> 0$. However, we show it is true for ‘almost all’ characters on the range $q^{1-o(1)} \leq H = o(q)$.

Using Pólya’s Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

We establish bounds for exponential sums twisted by generalized Möbius functions and their convolutions. As an application, we prove asymptotic formulas for certain weighted chromatic partitions by using the Hardy–Littlewood circle method. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.

We develop a generalisation of the square sieve of Heath-Brown and use it to give an alternate proof of one of the large sieve inequalities in our previous paper [‘A large sieve inequality for characters to quadratic moduli’, Preprint, https://web.maths.unsw.edu.au/~ccorrigan/preprint6.pdf].

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, Tr. Mat. Inst. Steklova 132 (1973), 247–250].

Lehmer [‘On certain character matrices’, Pacific J. Math. 6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math. 10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

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.

We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of the Fourier transform of the indicator function of the integer points on a hypersurface.

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.