On sum-product representations in $\\Bbb Z_q$

Given positive integers a1,. .. , a k , we prove that the set of primes p such that p ≡ 1 mod ai for i = 1,. .. , k admits asymptotic density relative to the set of all primes which is at least k i=1 1 − 1 ϕ(a i) , where ϕ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer n such that n ≡ 0 mod ai for i = 1,. .. , k admits asymptotic density which is at least k i=1

Acta Mathematica, 1998

Journal d'Analyse Mathématique, 2014

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q ′ = p|q p of the modulus q satisfies log q ′ ∼ log q. Some applications to zero free regions of Dirichlet L-functions and the Pólya and Vinogradov inequalities are indicated. In this paper we will discuss short character sums for moduli with small prime factors. In particular, we will revisit the arguments of Graham-Ringrose [GR] and Postnikov [P]. Our main result is an estimate valid for general moduli, which improves on the known estimates in certain situations. It is well known that non-trivial estimates on short character sums are important to many number theoretical issues. In particular, they are relevant in establishing density theorems for the corresponding Dirichlet L-functions. In the literature, several bounds on short incomplete character sums to some modulus q may be found, depending on the nature of q. Burgess' bound applies for moduli q that are cube free, provided the summation interval I has size N ≫ q 1 4 +ǫ . Assuming q has small prime factors, nontrivial estimates may * 2000 Mathematics Subject Classification. 11L40, 11M06.

Journal of the American Mathematical Society

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. Here d_K is a positive number depending only on K. Either of these new results implies another conjecture of Erdos and Graham, that if S is a finite set of moduli greater than N, with a choice for residue classes r(n) mod n for n in S which covers the integers, then the largest member of S cannot be O(N). We further obtain stronger forms of these results and establish other information, includ...

2008

Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can one represent every element of $\Z/pZ$ as a sum of some elements of $A$ ? (3) When can one represent every element of $\Z/pZ$ as a sum of $l$ elements of $A$ ?

Comptes Rendus Mathematique, 2003

Our first result is a 'sum-product' theorem for subsets A of the finite field F p , p prime, providing a lower bound on max(|A + A|, |A · A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group F

Acta Arithmetica, 2005

The Ramanujan Journal, 2005

Duke Mathematical Journal, 2007

In this paper we consider the problem of finding upperbounds on the minimum norm of representatives in residue classes in quotient O/I, where I is an integral ideal in the maximal order O of a number field K. In particular, we answer affirmatively a question of Konyagin and Shparlinski, stating that an upperbound o(N (I)) holds for most ideals I, denoting N (I) the norm of I. More precise statements are obtained, especially when I is prime. We use the method of exponential sums over multiplicative groups, exploiting essentially the new bounds obtained by the methods in [BC1] and [BC2]. Introduction. Let I be an integral ideal in an algebraic number field K. For a residue class α ∈ O/I, denote by N I (α) the minimal norm of all elements of α. Thus N I (α) = min x∈α |N (x)|. (0.1) Following [KS] (Chapter 9), we define further L(K, I) = max α∈(O/I) * N I (α).

2015

A set A = Ak,n ⇢ [n] [ {0} is said to be an additive k-basis if each element in {0, 1, . . . , kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function (n) ! 1, we say that A is said to be a truncated (n)-representative k-basis for [n] if for each j 2 [↵n, (k ↵)n] the number of ways that j can be represented as a k-sum of elements of Ak,n is ⇥( (n)). In this paper, we follow tradition and focus on the case (n) = log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n!1. This result is a finite version of a result proved by Erdős and extended by Erdős and Tetali.