On the Lebesgue measure of the expressible set of certain sequences

AbstractThe paper gives a condition for the expressible set of a sequence to have Lebesgue measure zero

A note on the transcendence of infinite products

This paper has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme ‘Research and Development for Innovations’ funded by Structural Funds of the European Union and state budget of the Czech Republic and by grants no. ME09017, P201/12/2351 and MSM 6198898701.The paper deals with several criteria for the transcendence of infinite products of the form ∏n=1∞[bnaan]/bnaan where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a n } n=1 ∞ and {b n } n=1 ∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).Web of Science62362361

Explicit irrationality measures for continued fractions

AbstractLet τ=[a0;a1,a2,…], a0∈N, an∈Z+, n∈Z+, be a simple continued fraction determined by an infinite integer sequence (an). We are interested in finding an effective irrationality measure as explicit as possible for the irrational number τ. In particular, our interest is focused on sequences (an) with an upper bound at most (ank), where a>1 and k>0. In addition to our main target, arithmetic of continued fractions, we shall pay special attention to studying the nature of the inverse function z(y) of y(z)=zlogz

A criterion for linear independence of infinite products

Abstract: Using an idea of Erdős the paper establishes a criterion for the linear independence of infinite products which consist of rational numbers. A criterion for irrationality is obtained as a consequence

Additive combinatorics and number theory

We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs.