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Infinite Patterns that can be Avoided by Measure
Author(s) -
Kolountzakis Mihail N.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003056
Subject(s) - mathematics , null set , lebesgue measure , measure (data warehouse) , universal set , conjecture , set (abstract data type) , discrete mathematics , σ finite measure , combinatorics , infinite set , finite set , set function , affine transformation , invariant (physics) , lebesgue integration , probability measure , pure mathematics , mathematical analysis , borel measure , computer science , database , mathematical physics , programming language
A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A . All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A , there is a set E of positive measure such that ( x + tA )⊆ E fails for almost all (Lebesgue) pairs ( x , t ). Also, the exceptional set of pairs ( x , t ) (for which ( x + tA )⊆ E ) can be taken to project to a null set on the t ‐axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale‐invariant way), then there is E ⊆ R of positive measure which contains no affine copy of A . 1991 Mathematics Subject Classification 28A12.