Approximation orders of the unit in the beta-dynamical systems

作者：Cao, C. -Y.; Chen, Y. -H.^*

来源：Acta Mathematica Hungarica, 2018, 154(1): 90-104.

DOI：10.1007/s10474-017-0776-5

• approximation order
• beta-dynamical system
• Hausdorff dimension

For any real number beta > 1, let S-n(beta) be the partial sum of the first n items of the beta-expansion of 1. It was known that the approximation order of 1 by S-n(beta) is beta(-n) for Lebesgue almost all beta > 1. We consider the size of the set of beta > 1 for which 1 can be approximated with the other orders beta(-phi(n)), where phi is a positive function defined on N. More precisely, the size of the sets
{beta is an element of B : lim sup(n ->infinity) log(beta)(1 - S-n(beta))/phi(n) = -1}
and
{beta is an element of B : lim inf(n ->infinity) log(beta)(1 - S-n(beta))/phi(n) = -1}
are determined, where B = {beta > 1 : beta is not a simple Parry number}.

• 出版日期2018-2
• 单位华中农业大学; 中南财经政法大学