Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s ...

We show that a C *-algebra A is nuclear iff there is a number α < 3 and a constant K such that, for any bounded homomorphism u : A → B(H), there is an isomorphism ξ : H → H satisfying ξ −1 ξ ≤ Ku α and such that ξ −1 u(.)ξ is a *-homomorphism. In other words, an infinite dimensional A is nuclear iff its length (in the sense of our previous work on the Kadison similarity problem) is equal to 2. ...

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that co (or any E.-space) is a T;space, i.e. cannot be realized as the quotient of a nonlocally convex F-space by a one-dimensional subspace.

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let f : X → X where X is a compact metric space and f is continuous, onto and piecewise one-to-one and Y := lim ←− (X, f). Then starting with a measure μ1 on the Borel sets B(X), we recursively construct a sequence of probability measures {μn}n=1 on B(X) satisfying ...

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number α by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved as well as establish its analog for the ultrametric Mahler measure.

2. Simultaneous similarity of tuples of matrices over C. Problem 1 is notoriously difficult. We show that for the local ring H0 this problem reduces to a Problem 2 for certain kind of matrices. We then discuss certain special cases of Problem 2 as simultaneous similarity of tuples of matrices to upper triangular and diagonal matrices. L-property of pairs of matrices, that discussed next, is clo...

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer’s conjecture and in at least one case that the converse is true as well...

Abstract. An elementary but useful fact is that the numerator of the difference of two consecutive Farey fractions is equal to one. For triples of consecutive fractions the numerators of the differences are well understood and have applications to several interesting problems. In this paper we investigate numerators of differences of fractions which are farther apart. We establish algebraic ide...

A description of the Poisson boundary of random walks on discrete subgroups of semi-simple Lie groups in terms of geometric boundaries of the corresponding Riemannian symmetric spaces is given. Let G be a discrete group, and { a probability measure on G. A function f on G is called-harmonic if f(g) = P f(gx) (x) 8 g 2 G. The Poisson boundary of the pair (G;) is the probability space (?;) with a...

Let α be a Zd–action (d ≥ 2) by automorphisms of a compact metric abelian group. For any non–linear shape I ⊂ Zd, there is an α with the property that I is a minimal mixing shape for α. The only implications of the form “I is a mixing shape for α =⇒ J is a mixing shape for α” are trivial ones for which I contains a translate of J . If all shapes are mixing for α, then α is mixing of all orders....