关于反平均数的2个上界

Two Upper Bounds for the Antiaverage Numbers

对于整数k,θ≥3,β≥1,称k个元素集合S为(k;β,θ)^(0)自由集,如果S的最小元素为0且它没有互不相同的元素a_(ij)∈S(1≤j≤θ使得∑_(j=1)^(θ-1)a_(ij)=βa_(iθ)成立,S的最大元素记为max(S).反平均数定义为λ(k;β,θ)=min{max(S):S是(k;β,θ)^(0)自由集}.给出反平均数λ(k;β,θ)的2个界.

For integers k,θ≥3 and,β≥1,we say a k-set S as a（k;β,θ）~0-free set if thesmallest number of S is 0,and S does not contain distinct elements aij∈S（1≤j≤9） suchthat∑j=1^θ=1 aij =βαiθ.The maximum number of S is denoted as max（S）.The generalizedantiaverage number is defined asλ（k;β,θ） = min{max（S）：S is（k;β,θ）^0-free set}.Twobounds of antiaverage numbers X（k;β,θ） are shown here.