On the size of the set A(A +  1)

Abstract

Let Fp be the field of a prime order p. For a subset \({A \subset F_p}\) we consider the product set A(A + 1). This set is an image of A × A under the polynomial mapping f(x, y) = xy + x : Fp × Fp → Fp. In the present note we show that if |A| < p1/2, then $$|A(A + 1)| \ge |A|^{106/105+o(1)}.$$ If |A| > p2/3, then we prove that $$|A(A + 1)| \gg \sqrt{p\, |A|}$$ 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 $$|A(A + 1)| \gg |A|^{5/4}.$$