On discrete values of bilinear forms

2015/12/08 by Alex Iosevich, Iosevich, Alex, Oliver Roche-Newton +3 · 1 citation
Mathematics · #11B75 #68R05 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:11B75 #msc:68R05

paper · pdf · doi:10.48550/arxiv.1512.02670

13pp

arxiv created 2015/12/08 · arxiv updated 2015/12/10

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 ω a non-degenerate skew-symmetric bilinear form in the plane F². We prove that for finite a point set P⊂ F²∖\0\, the set T_ω(P) of nonzero values of ω in P× P, if nonempty, has cardinality Ω(N^9/13). A presumably near-sharp estimate Ω(N/log N) was claimed in the abovemnetioned paper over the reals for a symmetric or skew-symmetric form ω. 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× A, where A is a set of at least two reals, we establish the following sum-product type estimates: |AA+ AA|= Ω(|A|^19/12), and |AA-AA|= Ω( \frac|A|^26/17log^2/17|A|).

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