Super-exponential condensation without exact overlaps

TL;DR

This paper constructs self-similar sets on the line that lack exponential separation and exact overlaps, demonstrating that Hochman's exponential separation condition is insufficient for a complete understanding of self-similar set dimensions.

Contribution

It introduces examples of self-similar sets without exponential separation or exact overlaps, challenging the adequacy of Hochman's exponential separation condition.

Findings

Existence of self-similar sets without exponential separation

These sets do not generate any exact overlaps

Hochman's exponential separation condition is too weak

Abstract

We exhibit self-similar sets on the line which are not exponentially separated and do not generate any exact overlaps. Our result shows that the exponential separation, introduced by Hochman in his groundbreaking theorem on the dimension of self-similar sets, is too weak to describe the full theory.