On the number of zeros of the function $\zeta(s)$ on “almost all” short intervals of the critical line

Suppose $\varepsilon>0$ is an arbitrarily small fixed number,
$$ Y\geqslant Y_0(\varepsilon)>0,\quad H=Y^\varepsilon,\quad Y_1=Y^{\frac{11}{12}+\varepsilon},\quad Y\leqslant T\leqslant Y+Y_1. $$

Consider the relation
$$ N_0(T+H)-N_0(T)\geqslant cH\ln T, $$
where $c=c(\varepsilon)>0$ is a constant depending only on $\varepsilon$, and let $E$ denote the set of those $T$ in the interval $Y\leqslant T\leqslant Y+Y_1$ for which this relation does not hold. It is shown that the measure of this set satisfies $\mu(E)\leqslant Y_1Y^{-0.5\,\varepsilon}$.
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