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arXiv · 2106.01612

Falconer type functions in three variables

Doowon Koh·Thang Pham·Chun-Yen Shen
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Abstract

Let $f\in \mathbb{R}[x, y, z]$ 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 $\mathbb{R}$. 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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Doowon Koh, Thang Pham, Chun-Yen Shen. 2021-06-03. Falconer type functions in three variables. https://arxiv.org/abs/2106.01612

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