{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:17:09Z","timestamp":1776795429667,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"280","license":[{"start":{"date-parts":[[2012,8,2]],"date-time":"2012-08-02T00:00:00Z","timestamp":1343865600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n An\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -ary\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -radius sequence is a finite sequence of elements taken from an alphabet of size\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that any two distinct elements of the alphabet occur within distance\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let\n \n \n \n \n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f_k(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the shortest length of any\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -radius sequence. We improve on earlier estimates for\n \n \n \n \n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f_k(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by using tilings and logarithms. The main result is that\n \n \n \n \n \n f<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n \n \u223c\n \n <\/mml:mo>\n \n 1<\/mml:mn>\n k<\/mml:mi>\n <\/mml:mfrac>\n \n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n <\/mml:mrow>\n <\/mml:mrow>\n f_k(n)\\sim \\frac {1}{k}\\binom {n}{2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as\n \n \n \n \n n<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n n\\rightarrow \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n whenever there exists a tiling of\n \n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \\mathbb {Z}^{\\pi (k)}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by a certain cluster of\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n hypercubes. In particular this result holds for infinitely many\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , including all\n \n \n \n \n k<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 194<\/mml:mn>\n <\/mml:mrow>\n k\\le 194<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and all\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k+1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n 2<\/mml:mn>\n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2k+1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is prime. For certain\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , in particular when\n \n \n \n \n 2<\/mml:mn>\n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2k+1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is prime, we get a sharper error term using the theory of logarithms.\n <\/p>","DOI":"10.1090\/s0025-5718-2011-02510-x","type":"journal-article","created":{"date-parts":[[2011,8,2]],"date-time":"2011-08-02T14:39:31Z","timestamp":1312295971000},"page":"2439-2459","source":"Crossref","is-referenced-by-count":14,"title":["Constructing \ud835\udc58-radius sequences"],"prefix":"10.1090","volume":"81","author":[{"given":"Simon","family":"Blackburn","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James","family":"McKee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,8,2]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-314X(83)90002-1","article-title":"On a problem of Oppenheim concerning \u201cfactorisatio numerorum\u201d","volume":"17","author":"Canfield, E. 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Olivier, PARI\/GP version 2.3.4, available from http:\/\/pari.math.u-bordeaux.fr\/."},{"key":"15","series-title":"Carus Mathematical Monographs","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.5948\/UPO9781614440246","volume-title":"Algebra and tiling","volume":"25","author":"Stein, Sherman K.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0883850281"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2012-81-280\/S0025-5718-2011-02510-X\/S0025-5718-2011-02510-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-280\/S0025-5718-2011-02510-X\/S0025-5718-2011-02510-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:21:13Z","timestamp":1776792073000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-280\/S0025-5718-2011-02510-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,8,2]]},"references-count":15,"journal-issue":{"issue":"280","published-print":{"date-parts":[[2012,10]]}},"alternative-id":["S0025-5718-2011-02510-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02510-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2011,8,2]]}}}