Asymptotic densities of sets of positive integers
Let be a finite field of characteristic . Let be the field of formal Laurent series in with coefficients in . That is,with and . We discuss the distribution of for , wheredenotes the nonnegative part of . This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic distribution for with for some . This distribution is...
Soit une suite strictement croissante d’entiers reconnaissable par un automate fini. Nous montrons qu’une condition nécessaire et suffisante pour que l’ensemble normal associé a soit exactement est que l’un au moins des sommets qui reconnaît la suite soit précédé dans le graphe de l’automate par un sommet possédant au moins deux circuits fermés distincts. Cette condition peut se traduire quantitativement en disant que la suite doit être plus “dense” que toute suite exponentielle.
Let be a self-similar set with similarities ratio and Hausdorff dimension , let be a probability vector. The Besicovitch-type subset of is defined aswhere is the indicator function of the set . Let and be a gauge function, then we prove in this paper:(i) If , thenmoreover both of and are finite positive;(ii) If is a positive probability vector other than , then the gauge functions can be partitioned as follows
It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.
We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition,...