Onthe gaps inmultiplicatively closed sets generated by atmost two elements

Proceedings of the Indian Academy of Sciences. Mathematical sciences

C P ANIL KUMAR^1¹ 

[1]No. 104, Bldg. 23, Lakshmi Paradise, 5th Main, 11th Cross, LN Puram, Bengaluru 560 021, India^1

关键词: Multiplicatively closed sets;    log-rationality;    gaps;    Weyl-equidistributive criterion;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We prove in the the main theorem, Theorem 3.2, that the multiplicatively closed subset of natural numbers,generated by two elements $1$ < $p_{1}$ < $p_{2}$ with $\alpha=\frac{\log\ p_{1}}{\log\ p_{2}}$ irrational, has arbitrarily largegaps by explicitly constructing large integer intervals, with known factorization for the endpoints in terms of generators $p_{1},p_{2}$ obtained from the stabilization sequence of the irrational $\alpha$ (Definition 3.1). Example 5.6 is also illustrated.In the Appendix, for a finitely generated multiplicatively closed subset of natural numbers, we mention another constructive proof (refer to Theorem A.1}) for arbitrarily large gap intervals, where the factorization of the right endpoint is not known in terms of generators unlikein the constructive proof of the main result. The suggested general Question 1.1 remains still open

【 授权许可】

CC BY