Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Falconer lattice sets and the Erdos similarity problem

A. Iosevich, A. Yavicoli

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}.

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}.

Paper Structure

This paper contains 4 sections, 18 theorems, 402 equations.

Table of Contents

  1. Introduction
  2. Modified Falconer construction
  3. Bourgain's result on sumsets is not enough to prove the Erdos conjecture for Cantor sets
  4. A large family of Cantor sets of logarithmic Hausdorff dimension zero containing a triple sumset

Key Result

Theorem 1.2

Let where $(M_i)_{i=1}^{\infty}$ is an increasing sequence of positive integers such that Let $\phi:[2,\infty)\to(0,\infty)$ be monotone increasing with $\phi(t)\to\infty$. For each $i$, define and Set If for all sufficiently large $i$, then there exists an infinite set such that Consequently, by Bourgain's theorem Bourgain2003, the set $E$ satisfies the conclusion of the Erdős similarity

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3: Structural dichotomy
  • Lemma 2.1: Branching of the lattice structure
  • proof
  • Lemma 2.2: The intersection is uncountable
  • proof
  • Lemma 2.3: No slowly decreasing sequences near any point of the intersection
  • proof
  • Lemma 2.4: There is a rapidly decreasing sequence in the intersection
  • ...and 28 more