Recently, there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers. Some of our results are presented in the much more general setting of trace functions.

We study sums of squares of integers except for a fixed one. For any nonnegative integer n, we find the minimum number of squares of integers except for n whose sums represent all positive integers that are represented by a sum of squares except for n. This problem could be considered as a generalisation of the result of Dubouis [‘Solution of a problem of J. Tannery’, Intermédiaire Math. 18 (1911), 55–56] for the case when $n=0$.

The descent method is one of the approaches to study the Brauer–Manin obstruction to the local–global principle and to weak approximation on varieties over number fields, by reducing the problem to ‘descent varieties’. In recent lecture notes by Wittenberg, he formulated a ‘descent conjecture’ for torsors under linear algebraic groups. The present article gives a proof of this conjecture in the case of connected groups, generalizing the toric case from the previous work of Harpaz–Wittenberg. As an application, we deduce directly from Sansuc’s work the theorem of Borovoi for homogeneous spaces of connected linear algebraic groups with connected stabilizers. We are also able to reduce the general case to the case of finite (étale) torsors. When the set of rational points is replaced by the Chow group of zero-cycles, an analogue of the above conjecture for arbitrary linear algebraic groups is proved.

Let A be a set of natural numbers. A set B of natural numbers is an additive complement of the set A if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. We establish that if $A=\{a_i: i\in \mathbb {N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb {N}$ and $\liminf _{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb {N}$ such that $B\cap A = \varnothing $ and B is a sparse additive complement of the set A.

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the Hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map. This result is applied to state new equivalence conditions for the Riemann hypothesis in a series of equalities.

Given r non-zero rational numbers $a_1, \ldots, a_r$ which are not $\pm1$, we complete, under Hypothesis H, a characterisation of the Schinzel–Wójcik r-rational tuples (i.e. r-tuples of rational numbers for which the Schinzel–Wójcik problem has an affirmative answer) which satisfy that the sum of the exponents of the positive elements $a_i$ in the representation of $-1$ in terms of the elements $a_i$ in the multiplicative group $\langle a_1,\dots, a_r\rangle\subset \mathbb{Q}^*$ is even whenever $-1 \in \langle a_1,\dots, a_r\rangle.$

For an integer $m>1$, we denote by $P(m)$ the largest prime divisor of m. We improve a result of Stewart [‘On the greatest and least prime factors of ${n!}+1$, II’, Publ. Math. Debrecen 65(3–4) (2004), 461–480] by showing that $\limsup _{n \rightarrow \infty } P({n!}+1)/n \geqslant 1+9\log 2$. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup _{n \rightarrow \infty } P({n!}+f(n))/n \geqslant 1+9\log 2$. This improves a result of Luca and Shparlinski [‘Prime divisors of shifted factorials’, Bull. Lond. Math. Soc. 37(6) (2005), 809–817]. These improvements stem from an additional combinatorial idea that builds upon the works mentioned above.

For each closed subtorus T of $(\mathbb{R}/\mathbb{Z})^n$, let D(T) denote the (infimal) $L^\infty$-distance from T to the point $(1/2,\ldots, 1/2)$. The nth Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by D(T) as T ranges over the 1-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture directly, we study the qualitative structure of the sets $\mathcal{S}(n)$ via their accumulation points. This project brings into the picture the analogues of $\mathcal{S}(n)$ where 1-dimensional subtori are replaced by k-dimensional subtori or k-dimensional subgroups.

We show the Harris–Viehmann conjecture under some Hodge–Newton reducibility condition for a generalisation of the diamond of a non-basic Rapoport–Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficients coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters in the $ {\mathrm {GL}}_2$-case.

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

A system of linear equations in $\mathbb {F}_p^n$ is Sidorenko if any subset of $\mathbb {F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty $. A system of linear equations is common if any two-colouring of $\mathbb {F}_p^n$ yields at least as many monochromatic solutions to the system of equations as a random 2-colouring, asymptotically as $n\to \infty $. Both classification problems remain wide open despite recent attention.

We show that a certain generic family of systems of two linear equations is not Sidorenko. In fact, we show that systems in this family are not locally Sidorenko, and that systems in this family which do not contain additive tuples are not weakly locally Sidorenko. This endeavour answers a conjecture and question of Kamčev–Liebenau–Morrison. Insofar as methods, we observe that the true complexity of a linear system is not maintained under Fourier inversion; our main novelty is the use of higher-order methods in the frequency space of systems which have complexity one. We also give a shorter proof of the recent result of Kamčev–Liebenau–Morrison and independently Versteegen that any linear system containing a four-term arithmetic progression is uncommon.

Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$. In this paper, we investigate the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.

A 2009 article of Allcock and Vaaler explored the $\mathbb {Q}$-vector space $\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$, showing how to represent it as part of a function space on the places of $\overline {\mathbb {Q}}$. We establish a representation theorem for the $\mathbb {R}$-vector space of $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {R}$, enabling us to classify extensions to $\mathcal {G}$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {Q}$, i.e., elements of the algebraic dual of $\mathcal {G}$. Our results make heavy use of Dirichlet’s S-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.

For any integer $t \geq 2$, we prove a local limit theorem (LLT) with an explicit convergence rate for the number of parts in a uniformly chosen t-regular partition. When $t = 2$, this recovers the LLT for partitions into distinct parts, as previously established in the work of Szekeres [‘Asymptotic distributions of the number and size of parts in unequal partitions’, Bull. Aust. Math. Soc. 36 (1987), 89–97].

We study the local theta correspondence for dual pairs of the form $\mathrm {Aut}(C)\times F_{4}$ over a p-adic field, where C is a composition algebra of dimension $2$ or $4$, by restricting the minimal representation of a group of type E. We investigate this restriction through the computation of maximal parabolic Jacquet modules and the Fourier–Jacobi functor.

As a consequence of our results, we prove a multiplicity one result for the $\mathrm {Spin}(9)$-invariant linear functionals of irreducible representations of $F_{4}$ and classify the $\mathrm {Spin}(9)$-distinguished representations.

We give a construction of integral local Shimura varieties which are formal schemes that generalise the well-known integral models of the Drinfeld p-adic upper half spaces. The construction applies to all classical groups, at least for odd p. These formal schemes also generalise the formal schemes defined by Rapoport-Zink via moduli of p-divisible groups, and are characterised purely in group-theoretic terms.

More precisely, for a local p-adic Shimura datum $(G, b, \mu)$ and a quasi-parahoric group scheme ${\mathcal {G}} $ for G, Scholze has defined a functor on perfectoid spaces which parametrises p-adic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over $O_{\breve E}$. Scholze-Weinstein proved this conjecture when $(G, b, \mu)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G, \mu)$ of abelian type when $p\neq 2$, and when $p=2$ and G is of type A or C. We also relate the generic fibre of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu , {\mathcal {G}})$.

In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with $\mathbf {Q}_p$-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of G-locally analytic vectors, $W^{\mathrm {la}}$, is useful because $W^{\mathrm {la}}$ can be studied via the Lie algebra $\mathrm {Lie}(G)$, which simplifies the action of G. Additionally, $W^{\mathrm {la}}$ often behaves as a decompletion of W, making it closer to an algebraic or geometric object.

This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for $\mathbf {Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and characteristic p.

Our main theorem shows that under certain conditions, the map $W \mapsto W^{\mathrm {la}}$ acts as a descent, and the derived locally analytic vectors $\mathrm {R}_{\mathrm {la}}^i(W)$ vanish for $i \geq 1$. This result extends Theorem C of [Por24], providing new tools for propagating information about locally analytic vectors from characteristic $0$ to characteristic p.

We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic $0$ methods, the introduction of an integral multivariable ring $\widetilde {\mathbf {A}}_{\mathrm {LT}}^{\dagger ,\mathrm {la}}$ in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring $\mathbf {{A}}_{\mathbf {Q}_p}$ from the theory of $(\varphi ,\Gamma )$-modules in terms of locally analytic vectors.

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Let K be an infinite field. If $\alpha $ and $\beta $ are algebraic and separable elements over K, then by the primitive element theorem, it is well known that $\alpha +u\beta $ is a primitive element for $K(\alpha , \beta )$ for all but finitely many elements $u\in K$. If we let

$$ \begin{align*}\xi_K(\alpha, \beta) = \{u\in K : K(\alpha, \beta) \ne K(\alpha+u\beta)\}\end{align*} $$

be the exceptional set, then by the primitive element theorem, $|\xi _K(\alpha , \beta )| < \infty $. Dubickas [‘An effective version of the primitive element theorem’, Indian J. Pure Appl. Math. 53(3) (2022), 720–726] estimated the size of this set when $K = \mathbb {Q}$. We take K to be a finite extension over $\mathbb {Q}$ or $\mathbb {Q}_p$, the field of p-adic numbers for some prime p, and estimate the size of the exceptional set.

In 1967, Klarner proposed a problem concerning the existence of reflecting n-queens configurations. The problem considers the feasibility of placing n mutually nonattacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ “reflecting strip” of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which n is it possible to pair up the integers 1 through n with the integers $n+1$ through $2n$ such that no two of the sums or differences of the n pairs of integers are the same. We prove the existence of reflecting n-queens configurations for all sufficiently large n, thereby resolving both Slater’s and Klarner’s questions for all but a finite number of integers.