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The Minkowski ?(x) function and Salem's problem. II

  • Giedrius Alkauskas
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Abstract

In 1943, R. Salem asked whether the Fourier-Stieltjes transform of the Minkowski question-mark function vanishes at infinity. This problem was answered affirmatively by Jordan and Sahlsten in 2016 as a consequence of their general results on Gibbs measures for the Gauss map. In this note we give a self-contained proof for ?(x). Several aspects of our argument are substantially different, making a very short proof possible in the ?(x) case. Moreover, we obtain an explicit polynomial decay estimate, with exponent at least 0.0472. We then prove the existence of the correlation dimension of the Minkowski measure, thereby improving the previous upper bound for the optimal Fourier decay exponent from 0.4373 to 0.4223.

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Cite

BibTeXRIS
BibTeX
@misc{alkauskas2026minkowski,
  title = {{The Minkowski $?(x)$ function and Salem's problem. II}},
  author = {Giedrius Alkauskas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39181},
  eprint = {2609.39181},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39181},
}
RIS
TY  - UNPB
TI  - The Minkowski ?(x) function and Salem's problem. II
AU  - Giedrius Alkauskas
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39181
UR  - https://arxiv.org/abs/2609.39181
AB  - In 1943, R. Salem asked whether the Fourier-Stieltjes transform of the Minkowski question-mark function vanishes at infinity. This problem was answered affirmatively by Jordan and Sahlsten in 2016 as a consequence of their general results on Gibbs measures for the Gauss map. In this note we give a self-contained proof for ?(x). Several aspects of our argument are substantially different, making a very short proof possible in the ?(x) case. Moreover, we obtain an explicit polynomial decay estimate, with exponent at least 0.0472. We then prove the existence of the correlation dimension of the Minkowski measure, thereby improving the previous upper bound for the optimal Fourier decay exponent from 0.4373 to 0.4223.
ER  -