Weyl's uniform distribution under periodic perturbation (Q2103488)

In the paper under review, the author examines classical uniform distribution theory when there is a periodic (or almost periodic) perturbation present. The general observation about density is that periodic or almost periodic perturbations never prevent density (mod 1). More precisely, the author shows that if \(\alpha\) is irrational and \(f_1, \dots, f_m\) are almost periodic functions, then the sequence with general term \(\alpha n+f_1(n)+\cdots+f_m(n)\) is dense (mod 1). He generalizes this result for the polynomial case, by showing that under same assumptions, if further \(P_0, P_1, \dots, P_m\) are polynomials without constant terms, and \(P_0\) has at least one irrational coefficient, then the sequence with general term \(P_0(n)+f_1(P_1(n))+\cdots+f_m(P_m(n))\) is dense (mod 1). The author also considers the vector case, and polynomial-vector case of the above result. Then, he examines the uniform distribution of the sequences that have been considered already by him in the paper. His work shows that when a periodic perturbation is present the situation is different, and although density is always true, but this is not the case for uniform distribution. Finally, the author changes his perspective; instead of considering modulo 1 values as fractional parts on the interval \([0, 1)\), he considers them on the torus \(\mathbb{R}/\mathbb{Z}\), i.e., when we identify the points 0 and 1 in \([0, 1]\). He shows that all of his results on the (mod 1) density and uniform distribution hold also on the torus without any change.