2026/07/31 by N. Mora Cuellar, A. Iosevich, N. Kulkarni +2
Mathematics · #math.CA #math.MG
arxiv created 2026/08/01 · arxiv updated 2026/08/04
We settle a major case in the two-set regime of the Erdős similarity conjecture: the sum of a geometric sequence and an arbitrary infinite set is never measure universal. Here a set E⊂\R is measure universal if every measurable set of positive Lebesgue measure contains an affine copy of E. More precisely, if A⊂\R is infinite, a≠ 0, and 0<|r|<1, then neither \arn:n≥ 1\+A \qquadnor \arn:n≥ 1\-A is measure universal. More generally, the same conclusion holds when the geometric sequence is replaced by any set containing a lacunary sequence (bn) with -log bn=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 S1,S2⊂\R contain lacunary subsequences and I(W),J(W) count their terms that are at least e-W, then S1+S2 and S1-S2 are not measure universal whenever \limsupW→∞(I(W)J(W))/(W)=∞. 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 α1,α2>0, the stretched-exponential sumset \2^-nα1\+\2^-nα2\ is not measure universal whenever 1/α1+1/α2>1; analogous conclusions hold for difference sets.