arXiv subjects
Publications and source records attributed to Alexander Fish.
We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set.
Jin proved that whenever $A$ and $B$ are sets of positive upper density in $\Z$, $A+B$ is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains $\Z^d$. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or -- depending on the notation -- "productsets") are piecewise Bohr, a result which for $G=\Z$ was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group $G$, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.
In this paper we generalize Wiener's characterization of continuous measures to compact homogenous manifolds. In particular, we give necessary and sufficient conditions on probability measures on compact semisimple Lie groups and nilmanifolds to be continuous. The methods use only simple properties of heat kernels.
In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $\R^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ \R^n$ so that the canonical projection onto $ \T^n $ is $ \eps$-dense.
Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of $D$-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.
We prove that if a Borel probability measure (μ) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (μ) is either Lebesgue or has finite support.
We define a random Liouville function (λ_Q) which depends on a random set (Q) of primes and prove that (A_Q = \{n \in \mathbb{N} | λ_Q(n) = -1 \}) is normal almost everywhere. This fact enables us to generate a family of normal sets such that the equation (xy =z) is not solvable inside them. Additionally we prove that equations (xy=z^2, x^2 + y^2 = square, x^2 - y^2 = square) are solvable in any normal set and for any equation (xy=cn^2) ((c > 1 ), is not a square) there exists a normal set (A_c) such that the equation is not solvable inside (A_c).