相关论文: An improved bound on the sum-product estimate in $…

This paper proves a bound for exponential sums over orbits of vectors in $\mathbb{F}_p^d$ under subgroups of $\rm{GL}_d(\mathbb{F}_p)$. The main tool is a classification theorem for approximate groups due to Gill, Helfgott, Pyber, and…

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that…

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\mathcal{O}_{\epsilon}(n^{3/5+\epsilon})$ solutions of…

Erd\H{o}s proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that…

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue…

The research problem in this work is the relaxation of maximizing non-negative submodular plus modular with the entire real number domain as its value range over a family of down-closed sets. We seek a feasible point $\mathbf{x}^*$ in the…

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are coprime to $p$…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$,…

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…

We prove that the sequence $n!\,(\bmod\,p)$ occupies at least $\sqrt{\frac{3}{2}N}$ residue classes in the short interval $H\le n \le H+N$ and $N\gg p^{\frac{1}{4}}$ improving previously known trivial bound $\sqrt{N}.$ In the other…

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo…

Let $p\geq3$ be a large prime and let $n(p)\geq2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll…

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results…

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log…

We prove, that the sequence $1!, 2!, 3!, \dots$ produces at least $(\sqrt{2} + o(1))\sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $\mathcal{I} \subseteq \{0, 1, \dots, p - 1\}$ of length $N > p^{7/8 +…

We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the…

Given two subsets $A, B \subseteq \mathbb{F}_p$ and a binary relation $\mathcal{R} \subseteq A \times B$, the restricted sumset of $A, B$ with respect to $\mathcal{R}$ is defined as $A +_{\mathcal{R}} B = \{ a+b \colon (a,b) \notin…