Exact overlaps conjecture

Let $\{\varphi_i(x)=r_i x+t_i\}_{i=1}^m$ be a finite self-similar iterated-function system on $\mathbb{R}$ , where $0<|r_i|<1$ , and let $(p_i)_{i=1}^m$ be positive weights with $\sum_i p_i=1$ . Let $\mu$ be the associated self-similar measure satisfying $\mu=\sum_{i=1}^m p_i(\varphi_i)_\ast\mu$ . If $\chi=-\sum_{i=1}^m p_i\log|r_i|$ is the Lyapunov exponent and $h_{\mathrm{RW}}$ is the asymptotic Shannon entropy of the random walk generated by the affine maps $\varphi_i$ , then the conjectured dimension formula is $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ . In the equivalent exact-overlaps formulation, any dimension loss beyond the value predicted without overlaps must be explained by exact overlaps between distinct finite compositions of the maps, meaning that $\varphi_{i_1}\circ\cdots\circ\varphi_{i_n}=\varphi_{j_1}\circ\cdots\circ\varphi_{j_k}$ for two distinct words.

References

Primary source

arXiv

Additional references

The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line — arXiv — Samuel Kittle, Constantin Kogler

Posted 1 hour ago · edited 1 hour ago

Progress summary

Refreshed 1 hour ago

Claimed progress

A new preprint claims to settle the one-dimensional case, but the higher-dimensional conjecture remains open and the claim has not been independently verified.

The conjecture predicts that dimension loss for self-similar measures is explained by exact overlaps between distinct iterated maps. Earlier work established strong one-dimensional partial results but not the unrestricted conjecture.

Known results

Hochman (2014): dimension drop implies superexponentially close cylinders; for algebraic parameters, either exact overlaps occur or the dimension is the generic value $\min\{1,s\}$ .
For systems on the real line with algebraic contraction ratios, the conjectured dimension formula is proved, with arbitrary translations.

October 2026 one-dimensional claim

Samuel Kittle and Constantin Kogler report a variance-based entropy inequality that establishes the conjectured formula for one-dimensional self-similar measures. This is a claimed advance in a specialist preprint and has not been independently verified; higher-dimensional cases are outside its scope.

Current status (as of October 2026): The one-dimensional case is claimed solved by a new preprint but remains unverified; higher-dimensional cases remain open.

Sources

No solutions have been posted yet.