A nilpotent Roth theorem

Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then exists in L ²-norm for any u, v∈L ^∞(X, ℬ, μ). We also show that for A∈ℬ with μ(A)>0 one has . By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that is, for some A∈ℬ, μ(A)>0, one may have μ(A∩T _-n A∩S ^{- n} A)=0 for all n∈ℕ. Finally, we show that when T and S generate a nilpotent group of class ≤c, in L ²(X) for all u, v∈L ^∞(X) if and only if T×S is ergodic on X×X and the group generated by T ^-1 S, T ^-2 S ²,..., T ^-c S ^c acts ergodically on X. Oblatum 19-V-2000 & 5-VII-2001?Published online: 12 October 2001     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

On Gittins’ index theorem in continuous time

We give a new and comparably short proof of Gittins’ index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

On the Hausdorff dimension of generalized Besicovitch-Eggleston sets of d-tuples of numbers

We give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.     

A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

A Fubini theorem

Let I₀ be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I₀ and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak-Ros?anowski-Shelah stating that I₀ on N² strongly violates the countable chain condition.     

Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.     

Measurability implies continuity for solutions of functional equations - even with few variables

Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.