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arXiv · 2610.10511

The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line

Samuel Kittle·Constantin Kogler
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Abstract

A generalization of the exact overlaps conjecture predicts a formula for the dimension of self-similar measures in terms of the random walk entropy and the Lyapunov exponent. We prove the exact overlaps conjecture and its generalized version in dimension one. The proof relies on a new entropy inequality quantifying the information lost under addition of independent random variables in terms of a quantity averaging variance across scales.

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Samuel Kittle, Constantin Kogler. 2026-10-07. The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line. https://arxiv.org/abs/2610.10511

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