Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider $$ V(N,x) = \int_0^1 \left( \sum_{n=1}^N I_{t,x} (X_n) \right)^2 - N^2 x^2 ~dt. $$ This is the number variance of $X_1, \dots, X_N$. Here the integral is taken with respect to the center of the interval, and it is understood that the interval "wraps around" modulo 1 when $t$ is close to 0 or 1.
Almost surely, $V(N,x)$ should be of order $Nx$ for a wide range of $x$. This can be seen for example as being the expected value of a suitable $\chi_1^2$ random variable.
Question: throughout which range (depending on $N$) is it true that almost surely $$ V(N,x) \sim Nx? $$ This is essentially a question about empirical processes, and I think Komlos-Major-Tusnady says that the desired asymptotics holds uniformly for all $x$ from $0$ up to roughly $1/\log x$. But what is the maximal possible range for $x$ here? I think one can check that $V(N,x) \sim Nx$ will not be true for very large $x$, such as $x = 1/100$, say, and if I didn't miscalculate then I think one could show that it fails for infinitely many $N$ for $x$ of size $\approx 1/ \log \log N$ (coming from fluctuations of sums of random variables as in the law of the iterated logarithm). But what is the optimal threshold for the maximal size of $x$? Is it somewhere around $1/\log N$, or around $1/\log \log N$, or somewhere inbetween? I would assume that this problem can be solved with some ingredients from the literature, but I couldn't locate anything.
