Falconer type functions in three variables

2021/06/03 by Koh, Doowon, Pham, Thang, Shen, Chun-Yen · 2 citations
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.01612

Abstract

Let f∈ ℝ[x, y, z] be a quadratic polynomial that depends on each variable and that does not have the form g(h(x)+k(y)+l(z)). Let A, B, C be compact sets in ℝ. Suppose that dim_H(A)+dim_H(B)+dim_H(C)>2, then we prove that the image set f(A, B, C) is of positive Lebesgue measure. Our proof is based on a result due to Eswarathasan, Iosevich, and Taylor (Advances in Mathematics, 2011), and a combinatorial argument from the finite field model.

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