{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:10:07Z","timestamp":1776791407344,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"270","license":[{"start":{"date-parts":[[2010,9,28]],"date-time":"2010-09-28T00:00:00Z","timestamp":1285632000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We study various discrepancies with arbitrary weights in the\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n L_2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm over domains whose dimension is proportional to\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We are mostly interested in large\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The exponent\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of discrepancy is defined as the smallest number for which there exists a positive number\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that for all\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n there exist\n \n \n \n \n C<\/mml:mi>\n \n \n \n \u03b5\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n C\\,\\varepsilon ^{-p}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n points with discrepancy at most\u00a0\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove that for the most standard case of discrepancy anchored at zero, the exponent is at most\n \n \n \n \n 1.41274<\/mml:mn>\n \n \u2026\n \n <\/mml:mo>\n \n <\/mml:mrow>\n 1.41274\\dots \\,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which slightly improves the previously known bound\n \n \n \n \n 1.47788<\/mml:mn>\n \n \u2026\n \n <\/mml:mo>\n \n <\/mml:mrow>\n 1.47788\\dots \\,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For discrepancy anchored at\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \\vec {\\alpha }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and for quadrant discrepancy at\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \\vec \\alpha<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we prove that the exponent is at most\n \n \n \n \n 1.31662<\/mml:mn>\n \n \u2026\n \n <\/mml:mo>\n \n <\/mml:mrow>\n 1.31662\\dots \\,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n =<\/mml:mo>\n [<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\vec \\alpha =[1\/2,\\dots ,1\/2]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For unanchored discrepancy we prove that the exponent is at most\n \n \n \n \n 1.27113<\/mml:mn>\n \n \u2026\n \n <\/mml:mo>\n \n <\/mml:mrow>\n 1.27113\\dots \\,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The previous bound was\n \n \n \n \n 1.28898<\/mml:mn>\n \n \u2026\n \n <\/mml:mo>\n \n <\/mml:mrow>\n 1.28898\\dots \\,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is known that for all these discrepancies the exponent is at least\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-09-02314-x","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"983-992","source":"Crossref","is-referenced-by-count":1,"title":["On the exponent of discrepancies"],"prefix":"10.1090","volume":"79","author":[{"given":"Grzegorz","family":"Wasilkowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henryk","family":"Wo\u017aniakowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2009,9,28]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/1990404","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn, N.","year":"1950","journal-title":"Trans. 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