In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844] made a seminal contribution by linking the improvability of Dirichlet’s theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B> 1$. We determine the Hausdorff dimension of the following set: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

Let $K = \mathbf {R}$ or $\mathbf {C}$. An n-element subset A of K is a $B_h$-set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A. Associated with the $B_h$-set $A = \{a_1,\ldots , a_n\}$ are the $B_h$-vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$. This article proves that “almost all” n-element subsets of K are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

In this paper, we investigate Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field with q elements, using Dwork’s method. Our study yields several arithmetic consequences. First, we establish that the zeta functions of a set of related affine varieties can reveal all Frobenius eigenvalues of the rigid cohomology of the variety up to a Tate twist. This result does not seem to be known for the $\ell$-adic cohomology. As a second application, we provide several q-divisibility lower bounds for the Frobenius eigenvalues of the rigid cohomology of the variety, in terms of the dimension and multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological degree.

Continuing our work on group-theoretic generalisations of the prime Ax–Katz Theorem, we give a lower bound on the p-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots,f_r)$ of r maps $f_j\,{:}\,A\rightarrow B_j$ between arbitrary finite commutative groups A and $B_j$ in terms of the invariant factors of $A, B_1,B_2, \cdots,B_r$ and the functional degrees of the maps $f_1,f_2, \dots,f_r$.

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

A real number is simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency at which the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on vertical arithmetic progressions. In particular, we show that

$$\begin{align*}\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \end{align*}$$

$2^{p/q}$

$2$

Weighted sieves are used to detect numbers with at most S prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilise the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes.

As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

Let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_n(\mathbb{A}_{\mathbb{Q}})$ with associated L-function $L(s, \pi)$. We study the behaviour of the partial Euler product of $L(s, \pi)$ at the centre of the critical strip. Under the assumption of the Generalised Riemann Hypothesis for $L(s, \pi)$ in conjunction with the Ramanujan–Petersson conjecture as necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point, which confirms a conjecture of Kurokawa (2012). As an application, we obtain results towards Chebyshev’s bias in the recently proposed framework of Aoki–Koyama (2023).

This paper discusses variants of Weber’s class number problem in the spirit of arithmetic topology to connect the results of Sinnott–Kisilevsky and Kionke. Let p be a prime number. We first prove the p-adic convergence of class numbers in a ${\mathbb{Z}_{p}}$-extension of a global field and a similar result in a ${\mathbb{Z}_{p}}$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the p-adic limit of the p-power-th cyclic resultants of a polynomial using roots of unity of orders prime to p, the p-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with p-adic limits being in ${\mathbb{Z}}$ and find the cases such that the base p-class numbers are small and $\nu$’s are arbitrarily large.

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

This article presents new rationality results for the ratios of critical values of Rankin–Selberg L-functions of $\mathrm {GL}(n) \times \mathrm {GL}(n')$ over a totally imaginary field $F.$ The proof is based on a cohomological interpretation of Langlands’s contant term theorem via rank-one Eisenstein cohomology for the group $\mathrm {GL}(N)/F,$ where $N = n+n'.$ The internal structure of the totally imaginary base field has a delicate effect on the Galois equivariance properties of the critical values.

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes $a_1$ and $a_2$ mod q, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.

Let $k{\geqslant} 2$ be an integer and let $\lambda$ be the Liouville function. Given k non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a non-trivial bound for the sums $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ under the existence of a Landau–Siegel zero for x in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau–Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis and Tao and Teräväinen.

We derive new cases of conjectures of Rubin and of Burns–Kurihara–Sano concerning derivatives of Dirichlet L-series at $s = 0$ in p-elementary abelian extensions of number fields for arbitrary prime numbers p. In naturally arising examples of such extensions one therefore obtains annihilators of class groups from S-truncated Dirichlet L-series for ‘large-enough’ sets of places S.

A square-full number is a positive integer for which all prime divisors divide the number at least twice. For a relatively prime pair $(a,b)\in \mathbb N\times \mathbb N\cup \{0\}$ and an affine polynomial $f(x)=ax+b$, the number of $n\leqslant N$ such that $f(n)$ is square-full is of order $N^{{1}/{2}}$. For $\varepsilon>0$ and an admissible quadratic polynomial $f(x)\in \mathbb Z[x]$, we show that the number of $n\leqslant N$ such that $f(n)$ is square-full is at most $O_{\varepsilon ,f}(N^{\varpi +\varepsilon })$ for some absolute constant $\varpi <1/2$. Under the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon ,f}(N^\varepsilon )$.

A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair $(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n, each term of which is congruent to P modulo $2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If $3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout’s identity.

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large C. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c \gt 0$. We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa’s question is equivalent to our arithmetic Ramsey question.

We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and k-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalisations to arbitrary one-dimensional patterns.