O(\sqrt{l_i})\right)\cdot (es/(k\gamma n))^{k\gamma n}\cdot 1/q(s)^n \cdot O(1)$ with (\[M1\] ) and Corollary \[N0\]. Observe that $l_i= \lambda_i k \gamma n, \sum_i \lambda_i=1, \sum \omega_i=1.$ Concerning $K(\bar{l})$ apply (\[K1\]).

For reals $a, b$ we let ${\cal U}_\varepsilon(a, b)=\{(c, d)| \, \, |c-a|, |d-b|<\varepsilon \}$ be the open square neighborhood of $(a, b).$ The notation $\bar{\lambda}, \bar{\omega}\in {\cal U}_\varepsilon(a, b)$ is used to mean $(\lambda_1, \lambda_2), (\omega_1, \omega_2)\in {\cal U}_\varepsilon(a, b).$ Theorem \[OPT\] is proved in Section \[PROOPT\].

\[OPT\] For any $\bar{\lambda}>0 $ there exist $\bar{a}, \bar{c}\,>\,0$ such that:\
(1) $\Psi( \bar{\omega}\, ,\, \bar{\lambda}\, ,\, \bar{a} \,,\, \bar{c}  \, )\, \le \, 3^{1-\gamma}. $\
(2) For any $\varepsilon>0, $ if $\bar{\lambda} \notin  {\cal U}_\varepsilon(1/3, 1/3)$ then $\Psi( \bar{\omega}\, , \bar{\lambda}\, , \bar{a} \,, \bar{c}   )\, \le \, 3^{1-\gamma} -  \delta  $ for a $\delta>0.$\

\[SUMOPT\] Let $U= {\cal U}_\varepsilon(1/3, 1/3)$ then $\sum_{\bar{\lambda} \notin U, \lambda_i>0 , \bar{\omega} } N(\bar{w}, \bar{l})/N_0  \,  <  \, C \cdot 3^{(1-\gamma)n}. $

The sum has only $O(n^4)$ terms. With Lemma \[LEMBA\] and Theorem \[OPT\] (2) we see that each term is bounded above by $ (3^{1-\gamma}- \delta)^n O(n)^{3/2}.$

To treat $(\lambda_1, \lambda_2) $ close to $(1/3, 1/3)$ we need a lemma analogous to Lemma \[MLOC\] for $K(\bar{l}).$ 