Far Eastern Mathematical Journal

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On the number of local minima of integer lattices


A. A. Illarionov, Y. A. Soyka

2011, issue 2, Ń. 149–154


Abstract
Let $E_s(N)$ be the average number of local minima of $s$-dimensional integer lattices with determinant equals $N$. We prove the following estimates
$$
\frac{2^{?1}}{(s?1)!}+O_s\left(\frac{1}{\ln N}\right) \le \frac{E_s(N)}{\ln^{s?1}N} \le \frac{2^s}{(s?1)!}+O_s\left(\frac{1}{\ln N}\right)
$$
\noindent for any prime N. Using this result we have a new lower bound for maximum number of local minima of integer lattices.

Keywords:
local minimum, multidimensional continuous fraction

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