<abstract xmlns="http://www.w3.org/1999/xhtml"> This paper deals with the distribution of αζn mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the distance to the nearest integer. We investigate the case of αζn all lying in prescribed small intervals modulo 1 for all large n, with focus on the case ‖αζn‖ ≤ ɛ for small ɛ > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small ɛ > 0 there are at most countably many values of α such that ‖αζn‖ ≤ ɛ for all large n, whereas larger ɛ induces an uncountable set. We investigate the value of ‖ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler’s 3/2-problem are implicitly deduced. We study similar questions for fixed α ≠ 0 as well. </abstract>

This paper deals with the distribution of αζn mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the distance to the nearest integer. We investigate the case of αζn all lying in prescribed small intervals modulo 1 for all large n, with focus on the case ‖αζn‖ ≤ ɛ for small ɛ > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small ɛ > 0 there are at most countably many values of α such that ‖αζn‖ ≤ ɛ for all large n, whereas larger ɛ induces an uncountable set. We investigate the value of ‖ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler’s 3/2-problem are implicitly deduced. We study similar questions for fixed α ≠ 0 as well.

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

This paper deals with the distribution of αζn mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the...

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 69 - 98

Received: Apr 17, 2015

Accepted: Mar 07, 2016

DOI: https://doi.org/10.1515/udt-2017-0005

Keywords
distribution modulo 1, distribution of powers, Pisot numbers, Salem numbers, Mahler’s 3/2-problem

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 69 - 98

Received: Apr 17, 2015

Accepted: Mar 07, 2016

DOI: https://doi.org/10.1515/udt-2017-0005

Keywordsdistribution modulo 1, distribution of powers, Pisot numbers, Salem numbers, Mahler’s 3/2-problem

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
distribution modulo 1, distribution of powers, Pisot numbers, Salem numbers, Mahler’s 3/2-problem