Related papers: Multilinear Exponential Sums With A General Class …

In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the…

This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.

We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation.…

For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer,…

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are…

We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse…

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not…

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which…

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and…

We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument…

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil,…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…

We present a hybrid approach to bounding exponential sums over kth powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents k of intermediate size.

This paper introduces a general technique for estimating the absolute value of pure Gaussian sums of order k over a prime p for a class of composite order k. The new estimate improves the classical estimate by a factor of about 2 or better…

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make…

In this paper, we use a recent method given by Rudnev, Shakan, and Shkredov (2018) to improve results on sum-product type problems due to Pham and Mojarrad (2018).

We study exponential sums of the form $\sum_{n=1}^N e^{2\pi i a b^n/m}$ for non-zero integers $a,b,m$. Classically, non-trivial bounds were known for $N\ge \sqrt{m}$ by Korobov, and this range has been extended significantly by Bourgain as…