Small Sets with Large Difference Sets

Abstract

For every ǫ > 0 and k ∈ N, Haight constructed a set A ⊂ ZN (ZN stands for the integers modulo N ) for a suitable N , such that A − A = ZN and |kA|< ǫN . Recently, Nathanson posed the problem of constructing sets A ⊂ ZN for given polynomials p and q, such that p(A) = ZN and |q(A)|< ǫN , where p(A) is the set {p(a1, a2, . . . , an): a1, a2, . . . , an ∈ A}, when p has n variables. In this paper, we give a partial answer to Nathanson's question. For every k ∈ N and ǫ > 0, we find a set A ⊂ ZN for suitable N , such that A − A = ZN , but |A 2 + kA|< ǫN , whereWe also extend this result to construct, for every k ∈ N and ǫ > 0, a set A ⊂ ZN for suitable N , such that A − A = ZN , but |3A 2 + kA|< ǫN , where 3A 2 + kA = {a1a2 + a3a4 + a5a6 + b1 + b2 + • • • + b k : a1, . . . , a6, b1, . . . , b k ∈ A}.