Construction of normal numbers via pseudo-polynomial prime sequences

Abstract

Abstract. In the present paper we construct normal numbers in base q by concatenating q-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.

Unexpected distribution phenomenon resulting from Cantor series expansions

“…Many similar and more sophisticated results have been proven since then. For example, J. Vandehey [45] and M. Madritsch and R. Tichy [26] have given similar constructions. A more extensive list of results can be found in Y. Bugeaud's book [9].…”

Dynamical Systems and Uniform Distribution of Sequences

“….. These constructions were extended to more general classes of functions g (replacing the polynomials) (see [11,17,18,22,23,29]) and the concatenation of [g(p)] q along prime numbers instead of the positive integers (see [10,18,19,24]).…”

Normality in Pisot numeration systems

“…Nakai and Shiokawa [20] also evaluated polynomials at primes, and Madritsch [15] showed that numbers generated by pseudo-polynomial sequences along the primes are normal. Further constructions of normal numbers in the spirit of Copeland and Erdős and Erdős and Davenport include [14] and [17].…”

“…This construction was extended to general integer-valued polynomials by Davenport and Erdős [7] and by Schiffer [23] and Nakai and Shiokawa [18], [19] to more general polynomial settings. Nakai and Shiokawa [20] also evaluated polynomials at primes, and Madritsch [15] showed that numbers generated by pseudo-polynomial sequences along the primes are normal. Further constructions of normal numbers in the spirit of Copeland and Erdős and Erdős and Davenport include [14] and [17].…”