Equal moments division of a set

Let $N_q^{*}(m)$ be the minimal positive integer $N$, for which there exists a splitting of the set $[0,N-1]$ into $q$ subsets, $S_0$, $S_1$, ..., $S_{q-1}$, whose first $m$ moments are equal. Similarly, let $m_q^{*}(N)$ be the maximal positive integer $m$, such that there exists a splitting of $[0,N-1]$ into $q$ subsets whose first $m$ moments are equal. For $q=2$, these functions were investigated by several authors, and the values of $N_2^{*}(m)$ and $m_2^{*}(N)$ have been found for $m\le8$ and $N\le167$, respectively. In this paper, we deal with the problem for any prime $q$. We demonstrate our methods by finding $m_3^*(N)$ for any $N<90$ and $N_3^*(m)$ for $m\le 6$.