Falconer lattice sets and the Erdos similarity problem

Abstract: We show that a family of extremely thin sets satisfy the Erdős similarity conjecture. These examples lie outside the range covered by recent work of Shmerkin and Yavicoli \cite{ShmerkinYavicoli2025}. As we shall see, they have small logarithmic dimension. They do not contain affine copies of slowly decaying sequences, so the result does not follow from earlier work of Falconer and Eigen \cite{Falconer1984,Eigen}. On the other hand, they do contain sequences of rapid decay, for which the conjecture is still open in general. Our argument is based on Falconer lattice sets and a theorem of Bourgain \cite{Bourgain2003}.

• The paper introduces a modified Falconer lattice construction that generates thin sets with robust additive branching to tackle the Erdős similarity problem.
• It establishes a structural dichotomy based on lattice scaling and neighborhood radius, proving criteria for set finiteness versus sustained uncountability.
• The study quantifies logarithmic packing and Hausdorff dimensions while contrasting the roles of rapidly versus slowly decaying sequences in pattern formation.

Falconer Lattice Sets and the Erdős Similarity Problem

Introduction and Context

The paper "Falconer lattice sets and the Erdős similarity problem" (2604.01493) addresses the longstanding Erdős similarity problem in geometric combinatorics, specifically by constructing new families of extremely thin sets that satisfy the conjecture via mechanisms distinct from previous regimes. The Erdős similarity problem, originating in the 1970s, asks: Given any infinite set $C \subset \mathbb{R}$, does there exist a measurable set of positive measure that does not contain any affine copy of $C$? Although progress has been made for sets of large (classical or logarithmic) dimension and specific rapidly or slowly decaying sequences, the underlying structure responsible for the similarity property, particularly in thin and structured sets, has remained underexplored.

Main Contributions

Novel Construction Using Modified Falconer Lattice Sets

The authors construct sets via a parameterized modification of Falconer's classical lattice-based sets. Let $M_i$ be a rapidly increasing sequence and $\phi$ a monotone function diverging to infinity, and consider the iterative lattice scales $q_1=2$, $q_{i+1}=q_i^{M_i}$. For each $i$, a neighborhood of the rescaled lattice $G_i = (1/q_i)[0, q_i] \cap \mathbb{Z}$ of radius $r_i = q_i^{-\phi(q_i)}$ is considered. The intersection $E = \bigcap_{i=1}^\infty E_i$ (with $C$0) yields the central example class.

A structural dichotomy is proven:

• If $C$1 for all sufficiently large $C$2, $C$3 is finite.
• If $C$4 for all sufficiently large $C$5, $C$6 is uncountable and exhibits strong additive branching: there exists an infinite set $C$7 with $C$8.

The result builds a connection between the scale of the lattice branching (determined by $C$9) and the size of the neighborhoods (determined by $M_i$0), driving whether the resulting set collapses or retains intricate additive structure.

Implications for the Erdős Similarity Problem

Leveraging a theorem by Bourgain, the existence of an infinite set $M_i$1 with $M_i$2 implies that $M_i$3 satisfies the conclusion of the Erdős similarity conjecture. This additive branching mechanism is independent of both the positive logarithmic dimension (as in [Shmerkin, Yavicoli 2025]) and slowly decaying sequence regimes (as in [Falconer 1984], [Eigen]). Unlike prior frameworks, the constructed sets $M_i$4 may have logarithmic packing and Hausdorff dimension strictly less than 2 and 1, respectively—thus, they lie outside the previously settled parameter regimes.

Distinction from Existing Results

• Lack of Slowly Decaying Sequences: The sets $M_i$5 exclude affine copies of slowly decaying sequences ($M_i$6), precluding use of Falconer's or Eigen's results. A local rigidity argument around any point in $M_i$7 is established to show this exclusion.
• Presence of Rapidly Decaying Sequences: $M_i$8 contains sequences with $M_i$9 (e.g., $\phi$0), a regime where the conjecture remains open in general.
• The sets constructed differ from those in recent work by [Shmerkin, Yavicoli 2025], which require positive logarithmic dimension, by accommodating examples of vanishing (even zero) logarithmic dimension.

Covering and Dimension Estimates

Under mild additional hypotheses on the growth of $\phi$1 and decay of $\phi$2, the logarithmic packing and Hausdorff dimensions of $\phi$3 are quantified. For certain parameter choices, one can achieve $\phi$4 and even $\phi$5. Explicit example sequences are given that witness these bounds.

Limitations of the Additive Branching Mechanism

The authors also clarify that Bourgain’s triple sumset criterion does not universally resolve the Erdős similarity problem for all thin sets, particularly Cantor-type constructions with additional algebraic constraints. For instance, they provide explicit Cantor sets of logarithmic dimension zero that are linearly independent over $\phi$6 and do not contain any triple sumset $\phi$7—thus, the additive mechanism has strict limitations regarding universality.

Existence of Cantor Sets of Logarithmic Dimension Zero with the Similarity Property

Additionally, the paper constructs further examples of Cantor-type sets (not driven by Falconer’s lattice mechanism) with $\phi$8, which nonetheless do contain a triple infinite sumset and thus satisfy the Erdős similarity conjecture. This demonstrates that positive logarithmic dimension is not a necessary condition for the conjecture to hold in the thin set regime.

Technical Results and Highlights

• Branching Lemma: Quantifies the number of descendants per lattice point that persist across scales, securing the robustness of the triple sum structure in the intersection set $\phi$9.
• Threshold Phenomenon: The relationship between $q_1=2$0 and $q_1=2$1 precisely controls the dichotomy between collapse and expansion of $q_1=2$2.
• Absence of Slow Decay: A rigorous argument excludes slowly decaying sequences from $q_1=2$3.
• Dimension Control: Through careful combinatorial and covering arguments, explicit dimension bounds are proven for logarithmic packing and Hausdorff dimension, including explicit constructions meeting these bounds.
• Non-Universality: The existence of linearly independent Cantor sets with vanishing logarithmic dimension that cannot support an infinite triple sumset is established, sharply characterizing the scope of Bourgain’s approach.

Implications and Future Directions

This work brings new insight into the Erdős similarity problem by successfully constructing and characterizing sets outside both the positive logarithmic dimension and slow sequence decay frameworks. The essential feature is the use of strong additive branching arising from nested lattice structures, which can force the presence of infinite triple sumsets in sets of arbitrarily thin fractal dimension.

The results suggest that the frontier for the similarity problem extends into the domain of very thin sets, where traditional methods relying on dimension or decay fail. The explicit separation of rapidly and slowly decaying sequence mechanisms prompts further investigation into finer structural invariants governing universality for affine patterns in thin sets.

Future research could entail:

• Precise characterization of minimal (logarithmic) dimension constraints compatible with the similarity property, or, conversely, minimality conditions for the additive branching construction.
• Extensions to higher dimensions or more general group actions.
• Identification of other structural properties (e.g., algebraic, topological, or additive) in thin sets that either support or obstruct universal affine copies.

Conclusion

The paper deepens understanding of the Erdős similarity problem by exhibiting a new class of sets, generated by modified Falconer lattice mechanisms, which satisfy the conjecture even in a regime of vanishing or zero logarithmic dimension. The results clarify the influence of nested additive structure and highlight boundaries of applicability for existing additive and dimensional techniques, representing a substantial advance in geometric combinatorics and the study of pattern avoidance in fractal sets.