The number of integers representable as a sum of two squares on small intervals

Let $M(m,h)$ denote the number of natural numbers in the interval $(m;m+h)$ which are representable as a sum of two squares. Under the condition $n>\ln^{42,5+\varepsilon}X$, $\varepsilon>0$, a best possible lower bound for $M(m,h)$ is established for almost all $m\leqslant X$ (for all but $o(X)$).
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