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On arithmetic sums of fractal sets in R d
Author(s) -
Feng DeJun,
Wu YuFeng
Publication year - 2021
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12422
Subject(s) - hyperplane , mathematics , affine transformation , fractal , arithmetic progression , affine arithmetic , arithmetic , integer (computer science) , combinatorics , conformal map , set (abstract data type) , discrete mathematics , pure mathematics , geometry , mathematical analysis , computer science , programming language
A compact set E ⊂ R dis said to be arithmetically thick if there exists a positive integer n so that the n ‐fold arithmetic sum of E has non‐empty interior. We prove the arithmetic thickness of E , if E is uniformly non‐flat, in the sense that there existsε 0 > 0 such that for x ∈ E and 0 < r ⩽ diam ( E ) , E ∩ B ( x , r ) never staysε 0 r ‐close to a hyperplane in R d . Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self‐similar sets, self‐conformal sets in R d (with d ⩾ 2 ) and self‐affine sets in R 2 that do not lie in a hyperplane, and certain self‐affine sets in R d (with d ⩾ 3 ) under specific assumptions.