| 方金辉.关于干扰序列的完备性[J].数学年刊A辑,2020,41(3):279~282 |
| 关于干扰序列的完备性 |
| Note on the Completeness of Perturbed Sequences |
| Received:February 04, 2019 |
| DOI：10.16205/j.cnki.cama.2020.0019 |
| 中文关键词: 完备性, 干扰序列, Hegyv' ari定理 |
| 英文关键词:Complete, Perturbed sequence, Hegyv´ari’s theorem |
| 基金项目:国家自然科学基金 (No.,11671211) |
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| 中文摘要: |
| 设S={ s_1, s_2, cdots }是正整数序列, alpha是正实数,令S_alpha={ lflooralpha s_1rfloor, lflooralpha s_2rfloor, cdots },其中lfloor xrfloor 指的是不超过x的最大整数. 此序列S_{alpha}可以看成是S的干扰序列.定义 U_S={ alpha mid alpha是实数且所有充分大的整数均可以表示为S_{alpha}中有限个互异项的和}.2013 年, 通过改进Hegyv' ari的结果, Chen和Fang证明了:若s_{n+1}0, 其中mu (U_S) 是U_S的Lebesgue测度. 本文得到了一个更强的结果. |
| 英文摘要: |
| Let S = {s2, s2, · · · } be a sequence of positive integers and α be a positive real number. Denote Sα by the sequence {?αs1?, ?αs2?, · · · }, where ?x? denotes the greatest integer not greater than x. This sequence Sα can be viewed as a perturbed sequence of S.Let US be the set of all positive real numbers α such that all sufficiently large integers can be representable as the finite sum of distinct terms of Sα. In 2013, by improving a result of Hegyv′ari, Chen and Fang proved that: if sn+1 < γsn for all sufficiently large integers n,where 1 < γ < 2, and US = ?, then μ(US) > 0, where μ(US) is the Lebesgue measure of US.This paper obtains a strong |
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