Grigori Kolesnik

Ergodic averaging sequences

We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers. The Birkhoff and Von Neumann ergodic theorems give conclusions about convergence of average measurements of systems when the measurements are made at integer times. We consider the case when the measurements are made at timesa(n) or ([a(n)]) where the functiona(x) is taken from a class of functions called a Hardy field, and we also assume that |a(x)| goes to infinity more slowly than some positive power ofx. A special, well-known Hardy field is Hardy’s class of logarithmico-exponential functions.The main theme of the paper is to point out that for a functiona(x) as described above, a complete characterization of the ergodic averaging behavior of the sequence ([a(n)]) is possible in terms of the distance ofa(x) from (certain) polynomials.

Distribution modulo 1 of some oscillating sequences

Sequences of the form (P(n)f(Q(n)))n=1∞,P andQ polynomials,f a “highly differentiable” periodic function, are considered. The results of [3] concerning the recurrence of this sequence to its value forn=0 are given a quantitative form. Density and uniform distribution modulo 1 are studied for specialQ’s.

Distribution Modulo 1 of Some Oscillating Sequences. III

AbstractFor some oscillating functions, such as

Regularity of patterns in the factorization of n

h\left( x \right) = x^\pi \log ^3 \times \cos \times

Joint distribution of completely q-additive functions in residue classes

, we consider the distribution properties modulo 1 (density, uniform distribution) of the sequence

Estimation of some exponential sums containing the fractional part function and some other “non-standard" exponential sums

h\left( n \right)