On the exponent of discrepancies

Abstract:

We study various discrepancies with arbitrary weights in the $L_2$ norm over domains whose dimension is proportional to $d$. We are mostly interested in large $d$. The exponent $p$ of discrepancy is defined as the smallest number for which there exists a positive number $C$ such that for all $d$ and $\varepsilon$ there exist $C \varepsilon ^{-p}$ points with discrepancy at most $\varepsilon$. We prove that for the most standard case of discrepancy anchored at zero, the exponent is at most $1.41274\dots$, which slightly improves the previously known bound $1.47788\dots$. For discrepancy anchored at $\vec {\alpha }$ and for quadrant discrepancy at $\vec \alpha$, we prove that the exponent is at most $1.31662\dots$ for $\vec \alpha =[1/2,\dots ,1/2]$. For unanchored discrepancy we prove that the exponent is at most $1.27113\dots$. The previous bound was $1.28898\dots$. It is known that for all these discrepancies the exponent is at least $1$.