Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-π,π]$ converges are precisely the Martin-Löf random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-Löf random reals.

翻译：在过去的十五年间，一项旨在建立算法随机性与分析学及遍历理论中几乎处处定理之间关系的研究计划逐步发展起来。在调和分析领域，Franklin、McNicholl 和 Rute 利用 Carleson 定理的一个有效版本刻画了 Schnorr 随机性。本文证明：对于可计算的 $1<p<\infty$，弱可计算 $L^p[-π,π]$ 空间向量 Fourier 级数收敛的实数点恰好是 Martin-Löf 随机实数。进一步，我们证明 $L^1(\mathbb{R})$-可计算函数的 Poisson 积分径向极限与函数值在 Schnorr 随机实数处完全一致，而弱 $L^1(\mathbb{R})$-可计算函数的 Poisson 积分径向极限与函数值在 Martin-Löf 随机实数处完全一致。