The Lind-Lehmer Constant for ${\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$

We show that the minimal positive logarithmic Lind-Mahler measure for a group of the form $G={\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$ with $\mathrm{\mid }G\mathrm{\mid }\ge 4$ is $\frac{1}{\mathrm{\mid }G\mathrm{\mid }}\log (\mathrm{\mid }G\mathrm{\mid }-1)\mathrm{.}$ We also show that for $G=\mathbb Z_2 \times \mathbb Z_{2^n}$with$n\geq 3$thisvalueis$\frac{1}{|G|} \log 9.$Previouslytheminimalmeasurewasonlyknownfor$2$-groupsoftheform$\mathbb Z_2^k$or$\mathbb Z_{2^k}.$