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

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

2008-11-26

arxiv.org/abs/0811.4206v3

Mathematics

Number Theory

Minor corrections

Scientific paper

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 $|A|p^{2/3},$ 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 $$ |A(A+1)|\gg |A|^{5/4}. $$

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