Related papers: Sum-product phenomena for Ahlfors-regular sets

We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erd\H{o}s function $A(n) =…

Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem.…

A partial fraction decomposition of the Fermi function resulting in a finite sum over simple poles is proposed. This allows for efficient calculations involving the Fermi function in various contexts of electronic structure or electron…

We raise a conjecture that asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due…

Partial sum rules are widely used in physics to separate low- and high-energy degrees of freedom of complex dynamical systems. Their application, though, is challenged in practice by the always finite spectrometer bandwidth and is often…

The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof of a growth result for $f(A)$ where $A$ has small additive doubling, and improve the bound by removing logarithmic factors.…

Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…

In this article we generalize Borel's classical approximation results for the regular continued fraction expansion to the alpha-Rosen fraction expansion, using a geometric method. We give a Haas-Series-type result about all possible good…

We prove a new result on multiple summing operators and among other applications, we provide a new extension of Littlewood's $4/3$ inequality to $m$-linear forms.

We give a simple proof of a well-known theorem of G\'al and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in G\'al's theorem,…

We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop…

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is…

Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite…

In the asymptotic analysis of regular sequences as defined by Allouche and Shallit, it is usually advisable to study their summatory function because the original sequence has a too fluctuating behaviour. It might be that the process of…

This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szeg\"o Theorem we prove sum rule identities in a very general form. Then, we apply the result…

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent $s>1$. As a corollary, we improve upon a recent result of Orponen, by showing that if $A$ is…

We obtain new bounds of exponential sums modulo a prime $p$ with binomials $ax^k + bx^n$. In particular, for $k=1$, we improve the bound of Karatsuba (1967) from $O(n^{1/4} p^{3/4})$ to $O\left(p^{3/4} + n^{1/3}p^{2/3}\right)$ for any $n$,…

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law…

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is…