A random pointwise ergodic theorem with Hardy field weights
A random pointwise ergodic theorem with Hardy field weights is a scholarly work, published in 2015 in ''Illinois Journal of Mathematics''. The main subjects of the publication include harmonic analysis, pointwise convergence, graph theory, prime number theorem, measure, pointwise operation, biological sequence, probability measure, discrete mathematics, ergodic theory, almost everywhere, mathematics, field, and combinatorics. The authors prove that, almost surely, for\nevery measure-preserving system $(X,T)$ and every $f \\in L^1(X)$ the modulated,\nrandom averages \\ \\frac{1}{N} \\sum_{n = 1}^N e(p(n)) T^{a_n(\\omega)} f\\\nconverge to $0$ pointwise almost everywhere.\n.