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Entropic rates from eigenfunction structure for × 2 × 3

  • Peter Burton
  • Kate Juschenko
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Abstract

In a companion article \cite{RJnodyn} the present authors proved a form of the Rudolph--Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a × 2 × 3-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's × 2 × 3 conjecture. We show that a near return of the coordinate function x ↦ e2πix to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of x ↦ 2x forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.

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Cite

BibTeXRIS
BibTeX
@misc{burton2026entropic,
  title = {{Entropic rates from eigenfunction structure for $\times 2 \times 3$}},
  author = {Peter Burton and Kate Juschenko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04745},
  eprint = {2610.04745},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04745},
}
RIS
TY  - UNPB
TI  - Entropic rates from eigenfunction structure for × 2 × 3
AU  - Peter Burton
AU  - Kate Juschenko
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04745
UR  - https://arxiv.org/abs/2610.04745
AB  - In a companion article \cite{RJnodyn} the present authors proved a form of the Rudolph--Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a × 2 × 3-invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's × 2 × 3 conjecture. We show that a near return of the coordinate function x ↦ e^2πix to itself under the semigroup generated by $2$ and $3$ forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of x ↦ 2x forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.
ER  -