Entropy of convolutions on the circle

Elon Lindenstrauss*, David Meiri, Yuval Peres

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Given ergodic p-invariant measures {μi} on the 1-torus double-struck T sign = double-struck R sign/double-struck Z sign, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution μ1 * ⋯ * μn converges to log p. We also prove a variant of this result for joinings of full entropy on double-struck T signdouble-struck N sign. In conjunction with a method of Host, this yields the following. Denote σq(cursive Greek chi) = qcursive Greek chi (mod 1). Then for every p-invariant ergodic μ with positive entropy, 1/N ΣN-1n=0 σcnμ converges weak* to Lebesgue measure as N → ∞, under a certain mild combinatorial condition on {ck}. (For instance, the condition is satisfied if p = 10 and ck = 2k + 6k or ck = 22k.) This extends a result of Johnson and Rudolph, who considered the sequence ck = qk when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorff dimension of sum sets: For any sequence {Si} of p-invariant closed subsets of double-struck T sign, if Σ dimH(Si)/| log dimH (Si)| = ∞, then dimH(S1 + ⋯ + Sn) → 1.

Original languageAmerican English
Pages (from-to)871-904
Number of pages34
JournalAnnals of Mathematics
Volume149
Issue number3
DOIs
StatePublished - May 1999

Keywords

  • Convolution
  • Entropy
  • Hausdorff dimension
  • Uniform distribution

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