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PreprintCC0arXiv

Isaac's Condition (TU) is false

  • Ralf Stephan
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Abstract

In 2006 Isaac reduced the simple normality to base $2$ of √s, for s not a perfect square, to a property of dyadic expansions that he called Condition~(TU): the upper frequency of the digit~$1$ in √ν, along any subsequence, depends only on the tail of the expansion of ν. We show that Condition~(TU) is false, already for the full sequence. We give two proofs. The first is elementary and turns Isaac's own reduction against the condition. The second uses the theorem of Kaufman and Mosquera--Shmerkin on the Fourier decay of smooth images of self-similar measures, and gives counterexamples for almost every point of a Cantor measure. All results are machine-verified in Lean~4; the second proof uses the two cited theorems as hypotheses.

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Cite

BibTeXRIS
BibTeX
@misc{stephan2026isaacs,
  title = {{Isaac's Condition (TU) is false}},
  author = {Ralf Stephan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21899},
  eprint = {2610.21899},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21899},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Isaac's Condition (TU) is false
AU  - Ralf Stephan
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21899
UR  - https://arxiv.org/abs/2610.21899
AB  - In 2006 Isaac reduced the simple normality to base $2$ of √s, for s not a perfect square, to a property of dyadic expansions that he called Condition~(TU): the upper frequency of the digit~$1$ in √ν, along any subsequence, depends only on the tail of the expansion of ν. We show that Condition~(TU) is false, already for the full sequence. We give two proofs. The first is elementary and turns Isaac's own reduction against the condition. The second uses the theorem of Kaufman and Mosquera--Shmerkin on the Fourier decay of smooth images of self-similar measures, and gives counterexamples for almost every point of a Cantor measure. All results are machine-verified in Lean~4; the second proof uses the two cited theorems as hypotheses.
ER  -