A 1-Dimensional Subset of the Reals That Intersects Each of Its Translates in at Most a Single Point

Abstract

We construct a compact subset of R with Hausdorff dimension 1 that intersects each of its non-identical translates in at most one point. Moreover, one can make the set to be linearly independent over the rationals.

Large sets avoiding patterns

“…Notice that it is possible to ask for sets E that avoid triangles that are not necessarily isosceles, for instance triangles where the sidelength ratio is a prescribed constant κ. The results in [10,14] and Theorem 1.1 all apply to give a set with the same Hausdorff dimension 1/2 as above not containing t 1 , t 2 , t 3 such that |γ(t 2 ) − γ(t 1 )| = κ|γ(t 3 ) − γ(t 1 )|. However, the Hausdorff dimension bound in Theorem 1.3 becomes worse as κ moves farther away from 1.…”

“…Construction of E. The construction is of Cantor type with a certain memoryretaining feature inspired by the constructions of Keleti [10,11]. This distinctive feature is the existence of an accompanying queue that is, on one hand, generated by the construction and on the other, contributes to it.…”

“…Nonexistence of patterns such as the one proved by Keleti [10] is the primary focus of this article. A main contribution of [10] is best described as a Cantor-type construction with memory, where selection of basic intervals at each stage is contingent on certain selections made at a much earlier step of the construction, so as to defy certain algebraic relations from taking place. This idea has been instrumental in a large body of subsequent work involving nonexistence of configurations.…”

“…In particular, such a set contains no nontrivial arithmetic progression, as can be seen by setting x 2 = x 3 . A counterpoint to [10] is a result of Laba and the second author [12], who have established existence of three-term progressions in special "random-like" subsets of R that support measures satisfying an appropriate ball condition and a Fourier decay estimate. Higher dimensional variants of this theme may be found in [3,9].…”

Large sets avoiding linear patterns

“…Keleti [8] also constructed a compact set E ⊆ R with full Hausdorff dimension, which does not contain points…”