Type: Article

On the size of the set A(A+1)

Moubariz Z. Garaev; Chun-Yen Shen;

Abstract:

Let F(p) be the field of a prime order p. For a subset A subset of F(p) we consider the product set A(A + 1). This set is an image of A x A under the polynomial mapping f(x, y) = xy + x : F(p) x F(p) -> F(p). In the present note we show that if vertical bar A vertical bar < p(1/2), then vertical bar A(A + 1)vertical bar = vertical bar A vertical bar(106/105+o(1).) If vertical bar A vertical bar > p(2/3), then we prove that vertical bar A(A + 1)vertical bar >> root p vertical bar A vertical bar and show that this is optimal in general settings bound up to the implied constant. We also estimate the cardinality of A(A + 1) when A is a subset of real numbers. We show that in this case one has the Elekes type bound vertical bar A(A + 1)vertical bar >> vertical bar A vertical bar(5/4).