https://scholars.lib.ntu.edu.tw/handle/123456789/638954
Title: | Falconer type functions in three variables | Authors: | Koh, Doowon Pham, Thang CHUN-YEN SHEN |
Keywords: | Expanding functions | Falconer distance problem | The regular value theorem | Issue Date: | 15-Feb-2024 | Journal Volume: | 286 | Journal Issue: | 4 | Source: | Journal of Functional Analysis | 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 dimH(A)+dimH(B)+dimH(C)>2, then we prove that the image set f(A,B,C) is of positive Lebesgue measure. This dimensional condition is best possible in general. 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. As a consequence, we improve a result of Iosevich and Liu (2016) on the Falconer distance problem for Cartesian product sets in three dimensions. |
URI: | https://scholars.lib.ntu.edu.tw/handle/123456789/638954 | ISSN: | 00221236 | DOI: | 10.1016/j.jfa.2023.110246 |
Appears in Collections: | 數學系 |
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