Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant ad...Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant advances in additive combinatorics concerning the size and structure of sumsets of finite integer sets.展开更多
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 No.12371003)。
文摘Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant advances in additive combinatorics concerning the size and structure of sumsets of finite integer sets.
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.
