关于干扰序列的完备性

Note on the Completeness of Perturbed Sequences

设S={s1,s2,…}是正整数序列,α是正实数,令Sα={「αs1」,「αs2」,…},其中x指的是不超过「x」的最大整数.此序列Sα可以看成是S的干扰序列.定义US={α|α是实数且所有充分大的整数均可以表示为Sα中有限个互异项的和}.2013年,通过改进Hegyvari的结果,Chen和Fang证明了:若sn+1<γsn对所有充分大的整数n均成立,其中1<γ<2,而且US≠?,则μ(US)>0,其中μ(US)是US的Lebesgue测度.本文得到了一个更强的结果.

Let S={s1,s2,···}be a sequence of positive integers andαbe a positive real number.Denote Sαby the sequence{「αs1」,「αs2」,…},where「x」denotes the greatest integer not greater than x.This sequence Sa can be viewed as a perturbed sequence of S.Let US be the set of all positive real numbers a such that all sufficiently large integers can be representable as the finite sum of distinct terms of Sa.In 2013,by improving a result of Hegyvari,Chen and Fang proved that:if sn+1<γsn for all sufficiently large integers n,where 1<γ<2,and US≠?,thenμ(US)>0,whereμ(US)is the Lebesgue measure of US.This paper obtains a stronger result.