Related papers: Improving the Burgess bound via Polya-Vinogradov

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…

We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$.…

We improve using elementary means an explicit bound on the divisor function due to Friedlander and Iwaniec. Consequently we modestly improve a result regarding a sieving inequality for Gaussian sequences.

We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.

We prove an averaged version of a claim suspected to be true by Alladi, Erd\"os, and Vaaler. Qualitatively, the result states that a divisor sum of a multiplicative function, which obeys certain size constraints, derives most of its value…

We obtain optimal lower bounds for moments of theta functions. On the other hand, we also get new upper bounds on individual theta values and moments of theta functions on average over primes. The upper bounds are based on bounds of…

Some new Frobenius norm bounds of the unique solution to certain structured Sylvester equation are derived. Based on the derived norm upper bounds, new multiplicative perturbation bounds are provided both for subunitary polar factors and…

Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies…

We consider the problem of $\Omega$ bounds for the partial sums of a modified character, \textit{i.e.}, a completely multiplicative function $f$ such that $f(p)=\chi(p)$ for all but a finite number of primes $p$, where $\chi$ is a primitive…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the…

We discuss a general approach producing quantitative bounds on the number of sign changes of the weighted sums $$\sum_{n\le x}f(n)w_n$$ where $f:\mathbb{N}\to \mathbb{R}$ is a family of multiplicative functions and $w_n\in\mathbb{R}$ are…

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…

We prove an extension of the Bourgain-Sarnak-Ziegler theorem and then apply it to bound certain polynomial exponential sums with modular coefficients.

We slightly improve Garaev's recent sum-product estimate largely by using Plunneke's inequality a little more efficiently. We improve the exponent from 15/14 to 14/13.

We present simple randomized and exchangeable improvements of Markov's inequality, as well as Chebyshev's inequality and Chernoff bounds. Our variants are never worse and typically strictly more powerful than the original inequalities. The…

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of…

We present some results on character degree sums in connection with certain characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.

We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we…