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PreprintCC BY 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{kittle2026exact,
  title = {{The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line}},
  author = {Samuel Kittle and Constantin Kogler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10511},
  eprint = {2610.10511},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10511},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line
AU  - Samuel Kittle
AU  - Constantin Kogler
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10511
UR  - https://arxiv.org/abs/2610.10511
AB  - 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.
ER  -