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 $e<sup>{-o(n)}$. 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 $e<sup>{-W}$, 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 ${2<sup>{-n}}+A$ 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.

• The paper demonstrates that sumsets from lacunary sequences and any infinite set are not measure universal, challenging longstanding aspects of the Erdős similarity problem.
• It introduces a finite-grid blocking principle paired with near-additive-energy estimates to provide uniform bounds on the separation of cross-sums at varying scales.
• The work recovers classical non-universality cases and extends results to asymmetric regimes, offering new insights for additive combinatorics and geometric measure theory.

Non-Universality of Sumsets of Lacunary Sequences and Arbitrary Sets

Introduction and Problem Context

The paper "Non-universality of sumsets of lacunary sequences and arbitrary sets" (2607.03584) advances the understanding of the universality properties of sumsets involving lacunary sequences and arbitrary sets with respect to the Lebesgue measure. The central concern is the Erdős similarity problem, specifically the characterization of subsets $E \subset \mathbb{R}$ that are measure universal, i.e., those for which every set of positive Lebesgue measure contains an affine copy of $E$.

While it is well-known that finite sets are measure universal by the Lebesgue density theorem, Erdős conjectured that no infinite set is measure universal. Proving this for various classes of infinite sets has been a longstanding challenge, with major progress made for sumsets of three or more infinite sets (Bourgain (2607.03584)) and certain sublacunary sequences, but the two-set regime, especially for lacunary sequences with rapid decay, remained largely unresolved.

Main Results

Finite-Grid Criteria and Additive-Energy

The authors employ and refine Kolountzakis’ finite-gap criterion, demonstrating that non-universality follows from the existence of arbitrarily large finite blocks at bounded scale with minimal gap at least $e^{-o(n)}$. This approach is grounded in a precise additive-combinatorial analysis, with the main technical innovation being a near-additive-energy estimate. This estimate provides a uniform bound on how the cross-sums of two lacunary sequences can cluster, ensuring that a positive proportion are well-separated uniformly at all scales.

Product and Asymmetric Regimes

The core theorem is as follows: for sets $S_1, S_2 \subset \mathbb{R}$ containing lacunary subsequences $(b_i)$ and $(d_j)$ with counting functions $I(W)$, $J(W)$ (number of points above $e^{-W}$), if

$\limsup_{W\to\infty}\frac{I(W)\,J(W)}{W} = \infty,$

then $E$0 and $E$1 are not measure universal. The critical aspect is the lack of any scale-separation or relative-decay assumption between the sequences; the result is fully quantitative in terms of the trade-off between the two counting functions.

At the endpoint—when one sequence is as dense as a lacunary sequence can possibly be, i.e., $E$2—the authors establish that $E$3 and $E$4 are not measure universal for any infinite $E$5. This is the first two-set non-universality result where one summand is entirely arbitrary. Moreover, the necessity for lacunarity is shown to be present on only one summand; for the other summand, a packing-function lower bound (i.e., metric entropy) suffices.

Recovery and Extension of Classical Regimes

The results recover classical non-universality instances for double geometric sumsets and sets with stretched-exponential decay. Furthermore, they strictly extend to asymmetric ranges in parameter space: for instance,

$E$6

is not universal whenever $E$7, covering many previously inaccessible cases.

Technical Approach and Methodology

The proofs systematically apply and unify the block and grid principles for constructing large separated finite sets at bounded scales, critically using counting function asymptotics to guarantee the existence of separated blocks with minimal gap $E$8. The key combinatorial estimates (near-additive-energy bounds) reveal that cross-sums (and cross-differences) of lacunary sequences avoid clustering excessively, even in the absence of scale-separation hypotheses.

The transition from counting function-based to packing number-based criteria enables the passage from sequence structure to metric mass, thereby subsuming previous structural requirements by a less restrictive quantitative metric condition.

The paper robustly delineates its results from both Bourgain’s three-set theorem and the classical finite-set/positivity dimension results, emphasizing the regime of sets with polynomially many points per scale—a region not covered by previous universal or non-universal classes.

Implications and Future Directions

The established non-universality in the two-set regime for the broad class of sumsets comprising a lacunary sequence and an arbitrary infinite set has strong implications for additive combinatorics and geometric measure theory. It provides new lower bounds on the complexity required for a set to be measure universal and tightens the landscape of the Erdős similarity problem. The methodology introduces a robust blocking and energy-calculus technique that can potentially be adapted to other problems regarding avoidance phenomena in measure and topology.

A prominent avenue for future research concerns the endpoint regimes where the current bounds ($E$9) may be further sharpened, possibly via structural/algebraic approaches rather than counting/energy mechanisms. The paper also highlights the remaining open case for single geometric sequences and their universality status.

Beyond analytic number theory, these results may inform the study of avoidance sets, self-similar structures, and universality phenomena in real analysis and theoretical computer science, especially in the context of measurable avoidance of structured patterns.

Conclusion

This work provides a definitive set of new non-universality theorems for two-set sumsets involving lacunary sequences and arbitrary sets, situated within the framework of the Erdős similarity problem. By developing a finite-grid blocking principle allied with sharp additive-energy estimates, the authors uniformly treat product regimes, asymmetric sumsets, and configurations with minimal algebraic structure on one component. These results close several open cases and open new paths for quantitative universality/avoidance analysis in additive and geometric measure theory.