The Erd\H{o}s Similarity Conjecture for Two-Fold Sumsets with a Geometric Summand

Abstract

We settle a major case in the two-set regime of the Erd\H{o}s similarity conjecture: the sum of a geometric sequence and an arbitrary infinite set is never measure universal. Here a set $E\subset\R$ is measure universal if every measurable set of positive Lebesgue measure contains an affine copy of $E$. More precisely, if $A\subset\R$ is infinite, $a\neq 0$, and $0<|r|<1$, then neither \[ \{ar^n:n\ge 1\}+A \qquad\text{nor}\qquad \{ar^n:n\ge 1\}-A \] is measure universal. More generally, the same conclusion holds when the geometric sequence is replaced by any set containing a lacunary sequence $(b_n)$ with $-\log b_n=O(n)$. Bourgain proved non-universality for sums of three arbitrary infinite sets, whereas the two-set regime is one of the principal remaining cases. Crucially, our conclusion applies to $\{2^{-n}\}+A$ for every infinite $A$, even though the non-universality of $\{2^{-n}\}$ itself remains open. The arbitrary-summand theorem is the maximal lacunary-density endpoint of a general counting-function trade-off. If $S_1,S_2\subset\R$ contain lacunary subsequences and $I(W),J(W)$ count their terms that are at least $e^{-W}$, then $S_1+S_2$ and $S_1-S_2$ are not measure universal whenever \[ \limsup_{W\to\infty}\frac{I(W)J(W)}{W}=\infty. \] No scale-separation or relative-decay assumption is required. The proof combines a finite-grid implementation of Kolountzakis' criterion with a near-additive-energy estimate controlling the clustering of lacunary cross-sums. A packing-number variant replaces lacunarity on one factor by a quantitative metric-mass condition. In particular, for $\alpha_1,\alpha_2>0$, the stretched-exponential sumset \[ \{2^{-n^{\alpha_1}}\}+\{2^{-n^{\alpha_2}}\} \] is not measure universal whenever $1/\alpha_1+1/\alpha_2>1$; analogous conclusions hold for difference sets.