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- W3211385329 abstract "Let $phi (x, y)colon mathbb {R}^{d}times mathbb {R}^{d}to mathbb {R}$ be a function. We say ϕ is a Mattila–Sjölin type function of index γ if γ is the smallest number satisfying the property that for any compact set $Esubset mathbb {R}^{d}$ , ϕ(E,E) has a non-empty interior whenever $dim _{H}(E)>gamma $ . The usual distance function, ϕ(x,y) = |x − y|, is conjectured to be a Mattila–Sjölin type function of index $frac {d}{2}$ . In the setting of finite fields $mathbb {F}_{q}$ , this definition is equivalent to the statement that $phi (E, E)=mathbb {F}_{q}$ whenever |E|≫ qγ. The main purpose of this paper is to prove the existence of such functions with index $frac {d}{2}$ in the vector space $mathbb {F}_{q}^{d}$ ." @default.
- W3211385329 created "2021-11-22" @default.
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- W3211385329 date "2021-11-09" @default.
- W3211385329 modified "2023-10-16" @default.
- W3211385329 title "Mattila–Sjölin Type Functions: A Finite Field Model" @default.
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- W3211385329 doi "https://doi.org/10.1007/s10013-021-00538-z" @default.
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