Arithmetic progressions with constant weight

Abstract

Let k⩽ n be two positive integers, and let F be a field with characteristic p. A sequence f:{1,…, n}→ F is called k-constant, if the sum of the values of f is the same for every arithmetic progression of length k in {1,…, n}. Let V( n, k, F) be the vector space of all k-constant sequences. The constant sequence is, trivially, k-constant, and thus dim V(n,k,F)⩾1 . Let m(k,F)= min n=k ∞ dim V(n,k,F) , and let c( k, F) be the smallest value of n for which dim V(n,k,F)=m(k,F) . We compute m( k, F) for all k and F and show that the value only depends on k and p and not on the actual field. In particular, we show that if p∤k (in particular, if p=0) then m( k, F)=1 (namely, when n is large enough, only constant functions are k constant). Otherwise, if k= p r t where r⩾1 is maximal, then m( k, F)= k− t. We also conjecture that c( k, F)=( k−1) t+ φ( t), unless p> t and p divides k, in which case c( k, F)=( k−1) p+1 (in case p∤k we put t= k), where φ( t) is Euler's function. We prove this conjecture in case t is a multiple of at most two distinct prime powers. Thus, in particular, we get that whenever k= q 1 s 1 q 2 s 2 where q 1, q 2 are distinct primes and p≠ q 1, q 2, then every k-constant sequence is constant if and only if n⩾ q 1 2 s 1 q 2 2 s 2 − q 1 s 1−1 q 2 s 2−1 ( q 1+ q 2−1). Finally, we establish an interesting connection between the conjecture regarding c( k, F) and a conjecture about the non-singularity of a certain (0,1)-matrix over the integers.