Arithmetic progressions with constant weight

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 kconstant 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 6 | 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 = pt 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 6 | 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 = q1 1 q s2 2 where q1, q2 are distinct primes and p 6= q1, q2, then every k-constant sequence is constant if and only if n ≥ q1 1 q 2s2 2 − q s1−1 1 q s2−1 2 (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.