Classification theorems for sumsets modulo a prime

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

An elementary construction is given of an infinite sequence of natural numbers, having at least two different decompositions as sum of primes and no prime number appears in more than one of them.

2014

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a similar result we have to take a sequence of length atleast $9p-8$ . In this paper we will show that if we add a condition on the chosen sequence, then we can get a good upper and a lower bound for which similar results hold.

Acta Mathematica Hungarica, 1993

Number Theory, 1987

The central problem in additive number theory is as follows: Let A be a set of nonnegative integers. Describe the set of integers that can be written as the sum of h elements of A, with repetitions allowed. This sumset is denoted by hA. If hA is the set of all nonnegative integers, then A is called a basis a grder 8. If hA contains all sufficiently large integers, then A is called an asvm~totip basis a orde1: 41. Most of classical additive number theory is the study of sumsets hA, where A is the set of squares (Lagrange's theorem), or the k-th powers (Waring's problem), or the polygonal numbers (Gauss's theorem for triangular numbers or Cauchy's theorem for polygonal numbers of any order), or the primes (Goldbach's conjecture). Shnirel'man [14] created a new field of research in additive number theory when he discovered a simple criterion that implies that a set A of nonnegative integers is a basis of order h for some h. Much recent work in additive number theory concerns general properties of additive bases of finite order. In this paper we discuss some unsolved problems about bases.

Journal of the London Mathematical Society, 2006

Our first result is a 'sum-product' theorem for subsets A of the finite field Fp, p prime, providing a lower bound on max(|A + A|, |A · A|). The second and main result provides new bounds on exponential sums

Journal of Number Theory, 2013

On the other hand, for every prime p there is a subset C ⊂ F p with |C| > p-C 2 log log p (log p) 1/2 p such that there are no A, B with these properties.

In this paper, the author presents a special polynomial function ( ) 1 ( ) ( ) a P r n n r r fx x r φ = = − ∏ , { } 1 ( ) , , a P r r rφ ∈ L , φ is Euler's function. 1 ( ) , , a P r rφ L are integer numbers relatively prime to the a P and n is odd integer, then he obtains its value congruent modulo a P where P and a denote an odd optional prime and a natural number respectively so ( 1) ( 1) ( ) ( 1) (mod ) a P n P n a f x x P − − ⎡ ⎤ ≡ + − ⎣ ⎦ , x Z ∈ and ( ) [ ] f x Z x ∈ .Here, the extended coefficients of the function, summation, and multiplication of all the members of the reduced residue system congruent modulo a P are also obtained. A special problem is proved by two methods: 1-multiplicative method 2-using groups and rings theory.

Journal of the European Mathematical Society, 2006

The purpose of this paper is to investigate efficient representations of the residue classes modulo q, by performing sum and product set operations starting from a given subset A of Z q. We consider the case of very small sets A and composite q for which not much seemed known (nontrivial results were recently obtained when q is prime or when log |A| ∼ log q). Roughly speaking we show that all residue classes are obtained from a k-fold sum of an r-fold product set of A, where r log q and log k log q, provided the residue sets π q (A) are large for all large divisors q of q. Even in the special case of prime modulus q, some results are new, when considering large but bounded sets A. It follows for instance from our estimates that one can obtain r as small as r ∼ log q/ log |A| with similar restriction on k, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for A = H a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on min a∈Z * q max x∈H ax/q are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of x m = a (mod q) to composite modulus q.

2007

This is the origin of paper ‘On a Question of Davenport and Lewis on Character Sums and Primitive Roots in Finite Fields’. There is still a little to be typed. Abstract Let A ⊂ Fp with |A| > p and |A + A| < C0|A|. We give explicit constants k = k(C0, ε) and κ = κ(C0, ε) such that |Ak| > κp. The tools we use are Garaev’s sum-product estimate, Freiman’s Theorem and a variant of Burgess’ method. As a by-product, we also obtain similar result for proper generalized progression in Fp.