Large sets avoiding affine copies of infinite sequences

Abstract: A conjecture of ErdH{o}s states that for any infinite set

A s u b s e t e q m a t h b b R

, there exists

E s u b s e t e q m a t h b b R

of positive Lebesgue measure that does not contain any nontrivial affine copy of

A

. The conjecture remains open for most fast-decaying sequences, including the geometric sequence

A = 2^{- k} : k g e q 1

. In this article, we consider infinite decreasing sequences

A = a_{k} : k g e q 1

in

m a t h b b R

that converge to zero at a prescribed rate; namely

l o g (a_{n} / a_{n + 1}) = e^{v a r p h i (n)}

, where

v a r p h i (n) / n o 0

as

n o i n f t y

. 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 Borel set

m a t h c a l O s u b s e t e q m a t h b b R

of Hausdorff dimension 1, but Lebesgue measure zero, that avoids all nontrivial affine copies of

A c u p 0

.

Cites Work