Extremal problems on nonaveraging and nondividing sets
1980; Mathematical Sciences Publishers; Volume: 91; Issue: 1 Linguagem: Inglês
10.2140/pjm.1980.91.1
ISSN1945-5844
Autores Tópico(s)Optimization and Variational Analysis
ResumoA set A of integers is said to be non-averaging if the arithmetic mean of two or more members of A is not in A. A is said to be non-dividing if no member divides the sum of two or more others.In this paper we investigate some of the many extremal problems which arise in connection with nonaveraging and non-dividing sets. = 0Let r = t(B -I) 2 and partition {1, 2, , n} into r sets S l9 S 2 , , S r where H. L. ABBOTT It will be useful to associate with a the lattice point (d o (a), d^a), , d t _i(α)) in E\ Note that the lattice points corresponding to numbers in S ά lie on a sphere of radius V j .Next partition Si into k = I* sets, two numbers a and b in S 3 -being placed in the same set if d^a) = d t (b) (modi) for i = 0, 1, , t -1.Thus {1, 2, , ri) has been partitioned into kr -tl\B -I) 2 = s setsSuppose that for some m, 2 ^ m ^ i, and some i, 1 ^ i <^ s, there are distinct numbers y θ9 y ly , y m in A t such that(3 ) Vo + yi+ -•• + y m -i
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