Abstract We show that for a Salem number $\beta $ of degree d , there exists positive constant $c(d)$ where ^m$ is Parry integers m natural density $\ge c(d)$ . Further, we $c(6)>1/2$ and discuss relation to the discretized rotation in dimension $4$

A generalization of de Bruijn digraphs is defined for Parry’s α-expansions and it is shown that the characteristic polynomial of these graphs is in principle that of α. With the help of this result we prove that certain functionals of α-expansions, e.g. the number of specific digital patterns, satisfy a central limit theorem, which is an extension of a result due to Drmota [3].

We present three simple ideas which enable to determine easily the number of return words in some infinite words. Using these ideas, we give a new and very short proof of the fact that each factor of an Arnoux-Rauzy word of order m has exactly m return words. We describe the structure of return words for the Thue-Morse sequence and focus on infinite words which are associated to simple Parry nu...

India traditionally is an agrarian economy, where even today 40% of the country’s GDP is derived from agriculture and agriculture products. Lately the Indian farmer community was not exposed to the use of ICT for its well-being, but since the Indian private sector has entered the arena, the acceptability of the latest innovations and tools by the farmer has increased. EID Parry, the group compa...

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine’s theorem and Jarńık’s theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that ...

We present three simple ideas which enable to determine easily the number of return words in some infinite words. Using these ideas, we give a new and very short proof of the fact that each factor of an Arnoux-Rauzy word of order m has exactly m return words. We describe the structure of return words for the Thue-Morse sequence and focus on infinite words which are associated to simple Parry nu...

the estimate of entropy (sample entropy), has been introduced by vasicek (1976), for the first time. in this paper, we provide an estimate of entropy of order statistics, that is the extention of the entropy estimate. then we present an application of the entropy estimate of order statistics as a test statistic for symmetry of distribution versus skewness. the proposed test has been compared wi...