Improved bounds on the set A(A+1)
Combinatorics
2012-08-06 v4
Abstract
For a subset A of a field F, write A(A + 1) for the set {a(b + 1):a,b\in A}. We establish new estimates on the size of A(A+1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show that A(A+1) is of cardinality at least C|A|^{57/56-o(1)} for some absolute constant C, so long as |A| < p^{1/2}. In the real case we show that the cardinality is at least C|A|^{24/19-o(1)}. These improve on the previously best-known exponents of 106/105-o(1) and 5/4 respectively.
Keywords
Cite
@article{arxiv.1205.3937,
title = {Improved bounds on the set A(A+1)},
author = {Timothy G. F. Jones and Oliver Roche-Newton},
journal= {arXiv preprint arXiv:1205.3937},
year = {2012}
}