Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate of the maximal value $a(N)$ such that for any such $\mu$ there exists a (closed, for concreteness) rectangle $R\subset Q$ (with sides parallel to the sides of $Q$) such that $|\mu(R)-\lambda(R)|\geqslant a(N)$? I guess that the product $N\cdot a(N)$ should tend to infinity for large $N$, but quite slowly.
$\begingroup$
$\endgroup$
3
asked May 22, 2019 at 7:17
-
3$\begingroup$ Just look at pdfs.semanticscholar.org/a39e/… and references therein, if necessary, to get an idea of the current state of art. In dimension 2, the problem is settled completely, but in higher dimensions there is a gap between the powers of $\log$ in the lower and the upper bounds. $\endgroup$– fedjaCommented May 22, 2019 at 7:44
-
$\begingroup$ What is meant by "always"? for every $\mu$? is my understanding correct: for given $\mu$, define $b_\mu=\sup_R|\mu(R)-\lambda(R)|$, and $a(N)$ is defined as $\inf_\mu b_\mu$? $\endgroup$– YCorCommented May 22, 2019 at 8:15
-
$\begingroup$ @YCor yes, your understanding is correct, and Fedja's comment actually contains the answer: $a(N)$ behaves as $\log(N)/N$. $\endgroup$– Fedor PetrovCommented May 22, 2019 at 8:44
Add a comment
|