Non-universality of sumsets of lacunary sequences and arbitrary sets

Abstract

A set $E\subset\R$ is measure universal if every set of positive Lebesgue measure contains an affine copy of $E$. By a theorem of Bourgain, a sum of three infinite sets is never measure universal, while the two-set regime is one of the central open cases of the Erdős similarity conjecture. We develop a finite-grid method for the two-set regime, based on Kolountzakis' finite-gap criterion: non-universality follows whenever one can construct arbitrarily large finite blocks at bounded scale with minimal gap at least $e^{-o(n)}$. Our first main result is phrased through counting functions. If $S_1,S_2$ contain lacunary subsequences whose counting functions $I(W)$, $J(W)$, defined as the numbers of terms above $e^{-W}$, satisfy $\limsup_{W\to\infty} I(W)J(W)/W=\infty$, then $S_1+S_2$ and $S_1-S_2$ are not measure universal. No scale-separation or relative-decay hypothesis relates the two sequences; their decay rates may trade off against each other. The key ingredient is a near-additive-energy estimate: the cross-sums of two lacunary sequences have uniformly controlled clustering, so a positive proportion of them are well separated at every scale. Our second main result is the endpoint of this trade-off, where one summand is as dense as a lacunary sequence can be. If $S$ contains a lacunary subsequence $(b_i)$ with $-\log b_i=O(i)$, for instance any geometric sequence, then $S+A$ and $S-A$ are not measure universal for every infinite set $A\subset\R$; in particular $\{2^{-n}\}+A$ is never measure universal. To our knowledge this is the first two-set non-universality theorem in which one summand is completely arbitrary. In fact, lacunarity is needed on only one factor: a packing-function variant, proved by a related near-energy estimate, asks only that the other factor carry enough metric mass, with no lacunarity or sequence structure required of it.