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Let $(\Omega, \A, \mu)$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\Omega$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\Omega$…

Let $\nu$ be a probability measure that is ergodic under the endomorphism $(\times p, \times p)$ of the torus $\mathbb{T}^2$, such that $\dim \pi \mu < \dim \mu$ for some non-principal projection $\pi$. We show that, if both $m\neq n$ are…

We show that L^2-bounded singular integral in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost…

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect…

We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of…

We compute $L^2$-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to…

We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…

The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex…

Motivated by LHC results, we modify the usual criterion for naturalness by ignoring the uncomputable power divergences. The Standard Model satisfies the modified criterion ('finite naturalness') for the measured values of its parameters.…

In the search for clues to the matter-antimatter puzzle, experiments with atoms or molecules play a particular role. These systems allow measurements with very high precision, as demonstrated by the unprecedented limits down to $10^{-30}$…

Let $Q$ denote the space of signed measures on the Borel $\sigma$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\|q\|=\sup|\int\phi dq|$, where the supremum is taken over all Lipschitz with constant 1 functions…

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we…

Symmetry is one of the most general and useful concepts in physics. A theory or a system that has a symmetry is fundamentally constrained by it. The same constraints do not apply when the symmetry is broken. The quantitative determination…

In this work we discuss Dirichlet products evaluated at Fibonacci numbers. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's totient function, an upper bound on the number of primitive…

For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable…

This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved…

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform…

We show that one can always identify a point on an algebraic variety $X$ uniquely with $\dim X +1$ generic linear measurements taken themselves from a variety under minimal assumptions. As illustrated by several examples the result is…

We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on…

In classical density (or density-functional) estimation, it is standard to assume that the underlying distribution has a density with respect to the Lebesgue measure. However, when the data distribution is a mixture of continuous and…