arXiv subjects
Publications and source records attributed to Yakov G. Sinai.
We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.
We present a limit theorem describing the behavior of a probabilistic model for square-free numbers. The limiting distribution has a density that comes from the Dickman-De Bruijn function and is constant on the interval $[0,1]$. We also provide estimates concerning the error term in the limit theorem.
In this paper we consider the recurrent equation $$Λ_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)Λ_{q}Λ_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $Λ_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $Λ_p$ with $Λ_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.
We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(α,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $α\in [0,1)$ is the rotation number and $(α, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(α,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.
In this paper we prove the following renewal-type limit theorem. Given an irrational $α$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $α$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.
The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by 2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)-Conjecture asks the same question, except for a subset of Pi of density 1. The Structure Theorem proven in \cite{sinai} shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x+1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x+1)-Map, and expand on the consequences derived in \cite{sinai}. The generalizations we consider are determined by positive coprime integers, d and g, with g > d >= 2, and a periodic function, h(x). The map T is defined by the formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer for which T(x) is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive diffusion constant. This shows that it is natural to expect that typical trajectores return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity otherwise.