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)=minn=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=prt 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 Eulerʹs 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=q1s1q2s2 where q1,q2 are distinct primes and p≠q1,q2, then every k-constant sequence is constant if and only if n⩾q12s1q22s2−q1s1−1q2s2−1(q1+q2−1). 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.
