On discrete values of bilinear forms

TL;DR

This paper corrects a previous claim about the size of the set of values of a bilinear form on finite point sets, establishing new lower bounds and discussing open problems in the area.

Contribution

It provides a corrected lower bound for the set of bilinear form values and introduces sum-product estimates for specific sets, highlighting unresolved issues in the field.

Findings

Lower bound of (N^{9/13}) for the set of bilinear form values

Sum-product estimates: |AA+AA|= |A|^{19/12} and |AA-AA|= |A|^{26/17}/( ext{log}^{2/17}|A|)

Discussion of the open problem regarding the ( ext{N}/ ext{log N}) estimate

Abstract

This paper is an erratum to our paper, entitled "On an application of Guth-Katz theorem", Math. Res. Lett. 18 (2011), no. 4, 691-697. Let $F$ be the real or complex field and $\omega$ a non-degenerate skew-symmetric bilinear form in the plane $F^2$. We prove that for finite a point set $P\subset F^2\setminus\{0\}$, the set $T_\omega(P)$ of nonzero values of $\omega$ in $P\times P$, if nonempty, has cardinality $\Omega(N^{9/13}).$ A presumably near-sharp estimate $\Omega(N/\log N)$ was claimed in the abovemnetioned paper over the reals for a symmetric or skew-symmetric form $\omega$. However, the set-up for the proof was flawed. We discuss why we believe that justifying this claim in full strength is a major open problem. In the special case when $P=A\times A$, where $A$ is a set of at least two reals, we establish the following sum-product type estimates: $$ |AA+ AA|= \Omega \left(|A|^{19/12}\right), $$ and $$|AA-AA|= \Omega\left( \frac{|A|^{26/17}}{\log^{2/17}|A|}\right).$$