Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference a...Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.展开更多
Let Z/m Z be the ring of residual classes modulo m,and let A and B be nonempty subsets of Z/m Z.In this paper,the authors give the structure of A and B for which|A+B|=|A|+|B|-1=m-2.
基金supported by National Natural Science Foundation of China(Grant Nos.12171243 and 11922113)the National Key Research and Development Program of China(Grant No.2021YFA1000700)。
文摘Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.
基金supported by the National Natural Science Foundation of China(Nos.12101007,12371003)the Natural Science Foundation of Anhui Province(No.2008085QA06)。
文摘Let Z/m Z be the ring of residual classes modulo m,and let A and B be nonempty subsets of Z/m Z.In this paper,the authors give the structure of A and B for which|A+B|=|A|+|B|-1=m-2.
