TWO GENERALIZED CONSTANTS RELATED TO ZERO-SUM PROBLEMS FOR TWO SPECIAL SETS

Abstract

Let n ∈ N and A ⊆ Zn be such that A is non–empty and does not contain 0. Adhikari et al proposed two generalized constants related to the zero-sum problem. One is DA(n), which denotes the least natural number k such that for any sequence (x1, · · · , xk) ∈ Z k, there exists a non-empty subsequence (xj1, · · · , xjl) and (a1, · · · , al) ∈ A l such that � l i=1 aixji ≡ 0 (mod n). The other is EA(n), defined as the smallest t ∈ N such that for all sequences (x1,..., xt) ∈ Z t, there exist indices j1,..., jn ∈ N, 1 ≤ j1 < · · · < jn ≤ t and (ϑ1, · · · , ϑn) ∈ An with �n ϑixji i=1 ≡ 0 (mod n). S. D. Adhikari et al proposed characterizing any other sets for which EA(n) = n + 1 or even those for which EA(n) = n + j for specific small values of j. In this paper we give two kinds of sets, calculate DA(n) and EA(n) for these sets, and partially solve Adhikari’s problem. 1