Light matrices of prime determinant

Abstract:

For $A = \left (a_{i,j}\right )$ a square integer matrix of prime determinant $p$, set \[ w(A)=\sum _{i,j}\left |a_{i,j}\right |.\] We are interested in the smallest possible value $w_p$ for $w(A),$ and we show that \[ \lim _{p\rightarrow \infty } w_p/\log _2(p)=5/2.\] We also show that $w_p \leq 2.5 \log _2(p)$ if and only if $p=2,7,13,37$ or a Fermat prime. Our results can also be interpreted as being about addition chains or about presentations of finite cyclic groups.