{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:53:32Z","timestamp":1776840812512,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"357","license":[{"start":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T00:00:00Z","timestamp":1768953600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["23K03210"],"award-info":[{"award-number":["23K03210"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.\n                  <\/p>","DOI":"10.1090\/mcom\/4051","type":"journal-article","created":{"date-parts":[[2024,12,3]],"date-time":"2024-12-03T12:36:41Z","timestamp":1733229401000},"page":"339-361","source":"Crossref","is-referenced-by-count":4,"title":["A randomized lattice rule without component-by-component construction"],"prefix":"10.1090","volume":"95","author":[{"given":"Takashi","family":"Goda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,1,21]]},"reference":[{"key":"1","first-page":"64","article-title":"Estimates in the mean of the remainder term of quadratic formulas","volume":"1","author":"Bahvalov, N. S.","year":"1961","journal-title":"\\v{Z}. Vy\\v{c}isl. Mat i Mat. 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