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 $\mu_{1/4}$, is a spectral measure with a spectrum $\Lambda_4$, called the canonical spectrum,\[ \Lambda_4 := \set{ \sum_{j=0}^{m-1}\varepsilon_j4^j: m\ge 1,\ \varepsilon_j\in\{0,1\} }. \] For $f\in L^1(\mu_{1/4})$, let $S_n f$ be the $n$-th partial sum of its Mock Fourier series with respect to $\Lambda_4$. We prove that the associated maximal operator $S^{\ast}f=\sup_n|S_n f|$ is of weak type $(1,1)$. Consequently, $S_n f\to f$ $\mu_{1/4}$-almost everywhere on $\supp(\mu_{1/4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.