On the exponent of discrepancies

We study various discrepancies with arbitrary weights in the L₂ 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 ε there exist Cε^-p points with discrepancy at most ε. We prove that for the most standard case of discrepancy anchored at zero, the exponent is at most 1.41274..., which slightly improves the previously known bound 1.47788.... For discrepancy anchored at α and for quadrant discrepancy at we prove that the exponent is at most 1.31662 ... for α = [1/2, ..., 1/2]. For unanchored discrepancy we prove that the exponent is at most 1.27113 .... The previous bound was 1.28898 .... It is known that for all these discrepancies the exponent is at least 1.