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

TL;DR

This paper investigates the size of the set A(A+1) over finite fields and real numbers, establishing new bounds that improve understanding of polynomial image sets in additive combinatorics.

Contribution

It provides new lower bounds for |A(A+1)| in finite fields for different subset sizes and extends the analysis to real numbers, improving existing bounds.

Findings

For |A|<p^{1/2}, |A(A+1)|≥|A|^{106/105+o(1)}.

For |A|>p^{2/3}, |A(A+1)| is at least on the order of √(p|A|).

Over real numbers, |A(A+1)|≥|A|^{5/4}.

Abstract

Let $F_p$ 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 $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show that if $|A|<p^{1/2},$ then $$ |A(A+1)|\ge |A|^{106/105+o(1)}.$$ If $|A|>p^{2/3},$ then we prove that $$|A(A+1)|\gg \sqrt{p |A|}$$ and show that this is the 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 $$ |A(A+1)|\gg |A|^{5/4}. $$