The exponent of discrepancy is at most 1.4778...

Abstract:

We study discrepancy with arbitrary weights in the $L_2$ norm over the $d$-dimensional unit cube. The exponent $p^*$ of discrepancy is defined as the smallest $p$ for which there exists a positive number $K$ such that for all $d$ and all $\varepsilon \le 1$ there exist $K\varepsilon ^{-p}$ points with discrepancy at most $\varepsilon$. It is well known that $p^*\in (1,2]$. We improve the upper bound by showing that \[ p^*\le 1.4778842.\] This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent $p^*$ is $2.454$.