Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure

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 $\{\frac{x-1}{4},\frac{x+1}{4}\}$ with equal weights, denoted by ${\mu }_{1\mathrm{/}4}$, is a spectral measure with a spectrum ${\mathrm{\Lambda }}_4$, called the canonical spectrum,$${\mathrm{\Lambda }}_4:=\{\sum _{j=0}^{m-1}{\epsilon }_j{4}^j:m\ge 1,{\epsilon }_j\in \{0,1\}\}\mathrm{.}$$ For $f\in L^1({\mu }_{1\mathrm{/}4})$, let $S_{n}f$ be the $n$-th partial sum of its Mock Fourier series with respect to ${\mathrm{\Lambda }}_4$. We prove that the associated maximal operator $S^{\ast }f={\sup }_n\mathrm{\mid }{S}_{n}f\mathrm{\mid }$ is of weak type $(1,1)$. Consequently, $S_{n}f\to f$ ${\mu }_{1\mathrm{/}4}$-almost everywhere on $\text{supp}({\mu }_{1\mathrm{/}4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.