{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:16Z","timestamp":1753893796566,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We determine the combinatorial discrepancy of the hypergraph ${\\cal H}$ of cartesian products of $d$ arithmetic progressions in the $[N]^d$\u2013lattice ($[N] = \\{0,1,\\ldots,N-1\\}$).  The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden's theorem, namely the Gallai-theorem (1933).  We solve the discrepancy problem for $d$\u2013dimensional arithmetic progressions by proving ${\\rm disc}({\\cal H}) = \\Theta(N^{d\/4})$ for every fixed integer $d \\ge 1$.  This extends the famous lower bound of $\\Omega(N^{1\/4})$ of Roth (1964) and the matching upper bound $O(N^{1\/4})$ of Matou\u0161ek and Spencer (1996) from $d=1$ to arbitrary, fixed $d$.  To establish the lower bound we use harmonic analysis on locally compact abelian groups. For the upper bound a product coloring arising from the theorem of Matou\u0161ek and Spencer is sufficient.  We also regard some special cases, e.g., symmetric arithmetic progressions and infinite arithmetic progressions.<\/jats:p>","DOI":"10.37236\/1758","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:23:52Z","timestamp":1578720232000},"source":"Crossref","is-referenced-by-count":4,"title":["Discrepancy of Cartesian Products of Arithmetic Progressions"],"prefix":"10.37236","volume":"11","author":[{"given":"Benjamin","family":"Doerr","sequence":"first","affiliation":[]},{"given":"Anand","family":"Srivastav","sequence":"additional","affiliation":[]},{"given":"Petra","family":"Wehr","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,1,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:05:09Z","timestamp":1579323909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1,2]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1758","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,1,2]]},"article-number":"R5"}}