Large subsets of Euclidean space avoiding infinite arithmetic progressions

Abstract

It is known that if a subset of R \mathbb {R} has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each λ \lambda in [ 0 , 1 ) [0,1) , we construct a subset of R \mathbb {R} that intersects every interval of unit length in a set of measure at least λ \lambda , but that does not contain any infinite arithmetic progression.