On the exponential sum-product problem
type: Item
instance of: scholarly article
title: On the exponential sum-product problem
zbMATH Open document ID: 1183.11077
DOI: 10.1016/S0019-3577(08)80006-7
author: Igor E. Shparlinski
published in: Indagationes Mathematicae. New Series
publication date: 2009-03-09T00:00:00Z
review text: Let \(p\) be a large prime and let \(g\) be an element of \(\mathbb F_p\) of a (large) multiplicative order \(T\). Given sets \(\mathcal A, \mathcal B \subset \mathbb Z_T^*\), consider the sets \[ \mathcal U = \{ g^{ab} : a \in \mathcal A, b \in \mathcal B \}, \quad \mathcal V = \{ g^{a} + g^{b} : a \in \mathcal A, b \in \mathcal B \}. \] In this paper, the author obtains non-trivial lower bounds for \(\mu = \max( \#\mathcal U, \#\mathcal V)\). Suppose that \(\#\mathcal B \leq \#\mathcal A \leq T^{1-\delta}\) for some \(\delta > 0\). When \(T \geq p^{1/2 + \varepsilon}\) for some fixed \(\varepsilon > 0\), the author uses an exponential sum estimate due to \textit{M. Z. Garaev} and \textit{A. A. Karatsuba} [Arch. Math. 87, No. 1, 33--40 (2006; Zbl 1175.11045)] to give an explicit lower bound for \(\mu\) in terms of \(p, T, \#\mathcal A\) and \(\#\mathcal B\). In particular, when \(\#\mathcal A\) and \(\#\mathcal B\) are approximately equal and \(T\) is close to \(p\), that bound is non-trivial provided that \(\#\mathcal A \geq T^{3/4 - \delta}\). For smaller values of \(T\), the author obtains another, more qualitative result. Drawing on work by \textit{J. Bourgain} [Geom. Funct. Anal. 15, No. 1, 1--34 (2005; Zbl 1102.11041)], he shows that if \(T \geq p^{\epsilon}\) for some fixed \(\varepsilon > 0\), one has \(\mu \geq \#\mathcal Ap^{\kappa}\) for some \(\kappa > 0\), provided that \(\#\mathcal A \geq T^{\eta + \delta}\) for some \(\eta \in (0, 1)\).
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 5526746
zbMATH Keywords: exponential sums
zbMATH Keywords: sum-product estimates
zbMATH Keywords: finite fields
reviewed by: Angel V. Kumchev
MaRDI profile type: Publication
cites work: Exponential sums over Mersenne numbers
cites work: Estimates on exponential sums related to the Diffie-Hellman distributions
cites work: Mordell’s exponential sum estimate revisited
cites work: Exponential sum estimates in finite commutative rings and applications
cites work: ESTIMATES FOR THE NUMBER OF SUMS AND PRODUCTS AND FOR EXPONENTIAL SUMS IN FIELDS OF PRIME ORDER
cites work: A sum-product estimate in finite fields, and applications
cites work: Some doubly exponential sums over Zm
cites work: Double exponential sums over thin sets
cites work: Q5461873
cites work: An Explicit Sum-Product Estimate in Fp
cites work: The sum-product estimate for large subsets of prime fields
cites work: New estimates of double trigonometric sums with exponential functions
cites work: Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture
cites work: Sum-product Estimates in Finite Fields via Kloosterman Sums
cites work: Q3400049
cites work: A slight improvement to Garaev's sum product estimate
cites work: On the elliptic curve analogue of the sum-product problem
full work available at URL: s0019-3577(08)80006-7
OpenAlex ID: W2092233710
Recommended article: Bounds on exponential sums over small multiplicative subgroups
Recommended article: On a variant of sum-product estimates and explicit exponential sum bounds in prime fields
Recommended article: ESTIMATES FOR THE NUMBER OF SUMS AND PRODUCTS AND FOR EXPONENTIAL SUMS IN FIELDS OF PRIME ORDER
Recommended article: On a new exponential sum
Recommended article: Sum-Product Theorems and Applications
Recommended article: Some remarks on multilinear exponential sums with an application
Recommended article: New bounds on exponential sums related to the Diffie-Hellman distributions
Recommended article: Q4397640
Recommended article: Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors
Recommended article: The finite Littlewood problem in \(\mathbb{F}_p\)
description: scientific article; zbMATH DE number 5526746