/ w_i = \lambda_ik \gamma /\omega_i$ and $c_0=1$ and $ R(c_1, c_2)= (\lambda_1k, \lambda_2k). $

[*Comment.*]{} Observe that $\lambda_ik \gamma /\omega_i\,\approx \, k\gamma, \, a_i\approx s, c_i \approx 1.$

Our restriction on $\bar{\omega}$ implies that ${n \choose \bar{w}}\le O(1/n)\prod_i (1/\omega_i)^{\omega_i n}$ (Stirling), giving us one $O(1/n).$ We get $\prod_i M(l_i, w_i)/N_0 \le \\
\prod_i \left( (l_i/(a_ie))^{l_i} q(a_i)^{w_i}  \right)\cdot (es/(k\gamma n))^{k\gamma n}\cdot 1/q(s)^n \cdot O(1)$ applying Corollary \[N0\] and Lemma \[MLOC\] for the $M(l_i, w_i).$ Concerning $K(\bar{l})$ apply Lemma \[KLOC\] which gives us a second factor $O(1/n).$ Otherwise the proof is as the proof of Lemma \[LEMBA\].

Lemma \[LOCMAX\] is proved in Section \[PROLAPLA\].

\[LOCMAX\] The function $ \Psi( \bar{\omega}\, ,\, \bar{\lambda}\, ,\, \bar{a} \,,\, \bar{c}  \, )$ with $a_i, c_i$ given by $  Q(a_i) = \lambda_ik\gamma /\omega_i$ and $c_0=1$ and $ R(c_1, c_2)= (\lambda_1k, \lambda_2k) $ has a local maximum with value $3^{1-\gamma}$ for $\lambda_i= \omega_i=1/3.$ In this case we get $a_i=s$ and $c_i=1.$

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

Let $\varepsilon>0$ be such that $\Psi(\bar{\omega}, \bar{\lambda}, \bar{a}, \bar{c}) \le 3^{1-\gamma} $ for $\bar{a}, \bar{c} $ as specified in Lemma 