LARGE SETS AVOIDING AFFINE COPIES OF INFINITE SEQUENCES

A conjecture of Erdos states that for any infinite set A subset of R, there exists a Borel set E subset of R of positive Lebesgue measure that does not contain any non-trivial affine copy of A. The conjecture remains open for most fast-decaying sequences, including the geometric sequence A = {2(-k) : k >= 1}. In this article, we consider infinite decreasing sequences A = {a(k) : k >= 1} in R that converge to zero at a prescribed rate; namely log(a(n)/a(n+1)) = e(phi(n)), where phi(n)/n -> 0 as n -> infinity. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence A, we construct a Cantor set K subset of [0, 1] with measure arbitrarily close to 1, such that the set of Erd.os points is an element of subset of K has Hasudorff dimension 1.