In the paper we discuss the following type congruences: where is a prime, , , and are various positive integers with , and . Given positive integers and , denote by the set of all primes such that the above congruence holds for every pair of integers . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets and inclusion relations between them for various values and . In particular, we prove that for all , and , and for...
In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers.
We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number in is univoque and self-sturmian if and only if the -expansion of is of the form , where is a characteristic...
We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic...