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Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure
Abstract
In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system {(x−1)/4,(x+1)/4} with equal weights, denoted by μ1/4, is a spectral measure with a spectrum Λ4, called the canonical spectrum,Λ4 := set ∑j=0m−1εj4j: m≥ 1, εj∈{0,1} . For f∈ L1(μ1/4), let Sn f be the n-th partial sum of its Mock Fourier series with respect to Λ4. We prove that the associated maximal operator S∗f=supn|Sn f| is of weak type (1,1). Consequently, Sn f→ f μ1/4-almost everywhere on supp(μ1/4). This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.
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BibTeX
@misc{tang2026almost,
title = {{Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure}},
author = {Min-Wei Tang and Zhi-Yi Wu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.27796},
eprint = {2607.27796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27796},
}
RIS
TY - UNPB
TI - Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure
AU - Min-Wei Tang
AU - Zhi-Yi Wu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2607.27796
UR - https://arxiv.org/abs/2607.27796
AB - In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system {(x−1)/4,(x+1)/4} with equal weights, denoted by μ_1/4, is a spectral measure with a spectrum Λ_4, called the canonical spectrum,Λ_4 := set ∑_j=0^m−1ε_j4^j: m≥ 1, ε_j∈{0,1} . For f∈ L^1(μ_1/4), let S_n f be the n-th partial sum of its Mock Fourier series with respect to Λ_4. We prove that the associated maximal operator S^∗f=sup_n|S_n f| is of weak type (1,1). Consequently, S_n f→ f μ_1/4-almost everywhere on supp(μ_1/4). This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.
ER -