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