Arithmetic progressions with constant weight (Q1586772)

Let \(F\) be a field of characteristic \(p\), and \(\mathcal F\) be a family of subsets of a set \(X\). A function \(f:X\to F\) is uniform on \(\mathcal F\) if there exists \(\alpha\in F\) such that the sum over all members of \(\mathcal F\) of the values of \(f\) is \(\alpha\). The vector space of all such uniform functions is denoted by \(V(X,{\mathcal F},F)\) and called the uniformity space of \((X,{\mathcal F})\) over \(F\); its dimension is called the uniformity dimension of \((X,{\mathcal F})\) over \(F\). After reviewing papers in the literature concerned with the determination of the uniformity spaces of specific combinatorial structures, e.g. [\textit{Y. Caro} and \textit{R. Yuster}, Discrete Math. 202, 1-19 (1999; Zbl 0932.05069), J. Graph Theory 29, 151-166 (1998; Zbl 0929.05060)], the author proceeds to consider the uniformity aspects of fixed length arithmetic progressions in sequences. Let \(k\leq n\) be positive integers. A sequence \(f:\{1,2,...,n\}\to F\) is \(k\)-constant if the sum of all values of \(f\) is the same for every arithmetic progression of length \(k\) in \(\{1,2,...,n\}\); \(V(n,k,F)\) denotes the vector space of all \(k\)-constant sequences; \(m(k,F)=\min\{\dim V(n,k,F):n=k,k+1,\dots\}\); \(c(k,F)=\min\{n|\dim V(n,k,F)=m(k,F)\}\). (He observes that \(k\)-constant arithmetic progressions in the field \(\mathbb{Z}_2\) are discussed in [\textit{N. Alon} and \textit{Y. Caro}, J. Graph Theory 17, 177-192 (1993; Zbl 0783.05074)]. For positive integers \(n,k\) with \(k|n\), the divisor matrix \(A_{n,k}\) is a \(\phi(k)\times n\) matrix whose rows correspond to the positive integers less than \(k\) and prime to \(k\); the \((i,j)\)th entry is 1 or 0 according as \(k\) does or does not divide \(i-j\). The primary divisor matrix \(A_n\) is an \(n\)-rowed matrix formed from all the rows of all \(A_{n,k}\) for which \(k|n\). Theorem 1.1: If \(p=0\) or \(\text{gcd}(p,k)=1\), then \(m(k,F)=1\). Otherwise, let \(k=p^rt\), where \(r\geq 1\) is maximal; then \(m(k,F)=k-t\). Theorem 1.2: If \(t=q_1^{r_1}q_r^{r_2}\) where \(r_1\) and \(r_2\) are non-negative integers and \(q_1\) and \(q_2\) are primes, then, if \(p<t\) or \(\text{gcd}(p,k)=1\), \(c(k,F)=(k-1)t+\varphi(t)\). (\(\varphi\) is the Euler totient function.) Otherwise \(c(k,F)=(k-1)p+1\). Theorem 1.2 is derived from the following theorems: Theorem 1.5: If \(n\) has at most two distinct prime factors, then \(A_n\) is non-singular over any field. Theorem 1.6: If \(A_t\) is non-singular over \(F\), then, if \(p<t\) or gcd\((p,k)=1\), \(c(k,F)=(k-1)t+\varphi(t)\); otherwise \(c(k,F)=(k-1)p+1\). Two conjectures are stated: Conjecture 1.3 (generalizing Theorem 1.2): For every positive integer \(k\), if \(p<t\) or \(\gcd(p,k)=1\), then \(c(k,F)=(k-1)t+\varphi(t)\); otherwise \(c(k,F)=(k-1)p+1\). Conjecture 1.4 (generalizing Theorem 1.5): \(\det(A_n)\in\{-1,1\}\).