h-重和集中的无穷项算术级数  

Infinite arithmetic progressions in h-fold sumsets

作　　者：陈永高 李亚莉 Yonggao Chen;Yali Li

机构地区：[1]南京师范大学数学科学学院,数学研究所,南京210023 [2]河南大学数学与统计学院,开封475001

出　　处：《中国科学:数学》2025年第4期779-790,共12页Scientia Sinica:Mathematica

基　　金：国家自然科学基金(批准号:12171243和11901156)资助项目。

摘　　要：设0<w<1,k是满足k-1<w^(-16)k的整数.用E_(w)表示满足以下条件的最小正整数d:对于下渐近密度为w的任意非负整数集合A,(k+1)A总包含公差至多为d的无穷项算术级数,其中(k+1)A表示A中k+1个元素(可以相同)的和组成的集合.Erd}os等(1988)证明了E_(w)6 k2-k,并提出如下问题:是否一定有E1=k=O(k)?最近,Chen和Li(2019)证明了:对充分小的正实数w,有E_(w)>k1:5.本文证明了:对充分小的正实数w,E_(w)至多有两个素因子,并且E_(w)>k1:9.本文也证明了:假设存在满足0<c<12(3-√5)的实数c,使得对每个充分大的x,区间(x-c√x;x)内都至少有一个素数,则对充分小的正实数w,E_(w)总为素数.本文还提出一些猜想和问题供进一步的研究.For any 0<w<1,let k be the integer with k^(-1)<w-16 k.Define E_(w)by the least positive integer d such that for every set A of nonnegative integers with the lower asymptotic density w,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.Erdos et al.(1988)proved that E_(w)6 k^(2)-k and asked if E_(1=k)=O(k).Recently,Chen and Li(2019)proved that E_(w)>k^(1:5) for all sufficiently small positive real numbers w.In this paper,we prove that E_(w)has at most two prime factors and E_(w)>k1:9 for all sufficiently small positive real numbers w.We also prove that E_(w)is prime for all sufficiently small positive real numbers w under the assumption that for some c with 0<c<1/2(3-p 5),the interval(x-c√x;x)contains at least one prime for every sufficiently large x.Several conjectures and a problem are proposed for further research.

关 键 词：算术级数 公差 H- 重和 

分 类 号：O156[理学—数学]