On the size of the set A(A + 1)
By Moubariz Z. Garaev, Chun-Yen Shen. 2008
50 citations recorded by the source corpus.
AbstractLet 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 $$|A(A + 1)| \gg |A|^{5/4}.$$
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/2e8db491a49560508e254e40609e10c953ceb5a2
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
Citations in the corpus
- An improved point‐line incidence bound over arbitrary fields (2016, methodology): “Our argument is again in the style of Elekes, and it was used over R by Garaev and Shen [12].”