Improved bounds on the set A(A+1)

TL;DR

This paper improves bounds on the size of the set A(A+1) for subsets A in finite fields and the real line, advancing previous exponents with new estimates.

Contribution

The paper provides new lower bounds on the size of A(A+1) in finite fields and real numbers, surpassing previous known exponents.

Findings

Finite field bound: |A(A+1)| ≥ C|A|^{57/56-o(1)} for |A|< p^{1/2}

Real line bound: |A(A+1)| ≥ C|A|^{24/19-o(1)}

Improved exponents over previous bounds 106/105-o(1) and 5/4

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.