|
Let C n be a finite cyclic group of order n ≥ 2 . Every sequence S over C n can be written in the form S = ( n 1 g ) , … , ( n l g ) where g ∈ C n and n 1 , … , n l ∈ [ 1 , ord ( g ) ] , and the index ind ( S ) of S is defined as the minimum of ( n 1 + ⋯ + n l ) / ord ( g ) over all g ∈ C n with ord ( g ) = n . Let d 1 and r ≥ 1 be any fixed integers. We prove that, for every sufficiently large integer n divisible by d , there exists a sequence S over C n of length | S | ≥ n + n / d + O ( n ) having no subsequence T of index ind ( T ) ∈ [ 1 , r ] , which has substantially improved the previous results in this direction.
|
