{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:05:21Z","timestamp":1776837921848,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"347","license":[{"start":{"date-parts":[[2024,10,25]],"date-time":"2024-10-25T00:00:00Z","timestamp":1729814400000},"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":"<p>\n                    Given a matrix\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A element-of normal upper G normal upper L Subscript d Baseline left-parenthesis double-struck upper Z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">G<\/mml:mi>\n                                <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A\\in \\mathrm {GL}_d(\\mathbb {Z})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We study the pseudorandomness of vectors\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold u Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">u<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {u}_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    generated by a linear recurrence relation of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold u Subscript n plus 1 Baseline identical-to upper A bold u Subscript n Baseline left-parenthesis mod p Superscript t Baseline right-parenthesis comma n equals 0 comma 1 comma ellipsis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">u<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">u<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mspace width=\"0.667em\"\/>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>mod<\/mml:mi>\n                            <mml:mspace width=\"0.333em\"\/>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"2em\"\/>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\mathbf {u}_{n+1} \\equiv A \\mathbf {u}_n \\pmod {p^t}, \\qquad n = 0, 1, \\ldots , \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Superscript t\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">p^t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with a fixed prime\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and sufficiently large integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t greater-than-or-slanted-equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>\n                              \u2a7e\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t \\geqslant 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N.\u00a0M.\u00a0Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp.\u00a0654\u2013670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K.\u00a0Ford [Proc. London Math. Soc. (3) 85 (2002), pp.\u00a0565\u2013633]. This is combined with some ideas from the work of I.\u00a0E.\u00a0Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74\u201385 (in Russian)] which allows us to construct polynomial representations of the coordinates of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold u Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">u<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {u}_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and control the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic orders of their coefficients in polynomial representations.\n                  <\/p>","DOI":"10.1090\/mcom\/3895","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:50:58Z","timestamp":1698227458000},"page":"1355-1370","source":"Crossref","is-referenced-by-count":2,"title":["Distribution of recursive matrix pseudorandom number generator modulo prime powers"],"prefix":"10.1090","volume":"93","author":[{"given":"L\u00e1szl\u00f3","family":"M\u00e9rai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Igor","family":"Shparlinski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,10,25]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1090\/S0894-0347-05-00476-5","article-title":"Mordell\u2019s exponential sum estimate revisited","volume":"18","author":"Bourgain, J.","year":"2005","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/978-3-540-72053-9_5","article-title":"A remark on quantum ergodicity for CAT maps","author":"Bourgain, J.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540720522"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"1477","DOI":"10.1007\/s00039-008-0691-6","article-title":"Multilinear exponential sums in prime fields under optimal entropy condition on the sources","volume":"18","author":"Bourgain, Jean","year":"2009","journal-title":"Geom. Funct. 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Iberoam.","ISSN":"https:\/\/id.crossref.org\/issn\/0213-2230","issn-type":"print"},{"key":"12","series-title":"Scriptum, no. 5","volume-title":"Some theorems on Diophantine inequalities","author":"Koksma, J. F.","year":"1950"},{"key":"13","first-page":"654","article-title":"The distribution of digits in periodic fractions","volume":"89(131)","author":"Korobov, N. M.","year":"1972","journal-title":"Mat. Sb. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s002200100501","article-title":"On quantum ergodicity for linear maps of the torus","volume":"222","author":"Kurlberg, P\u00e4r","year":"2001","journal-title":"Comm. Math. 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Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/2522-0144","issn-type":"print"},{"key":"17","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random number generation and quasi-Monte Carlo methods","volume":"63","author":"Niederreiter, Harald","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712955"},{"issue":"16","key":"18","doi-asserted-by":"publisher","first-page":"14196","DOI":"10.1093\/imrn\/rnac226","article-title":"Equations and character sums with matrix powers, Kloosterman sums over small subgroups, and quantum ergodicity","author":"Ostafe, Alina","year":"2023","journal-title":"Int. Math. Res. Not. IMRN","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"key":"19","isbn-type":"print","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/978-1-4020-5404-4_15","article-title":"The arithmetic theory of quantum maps","author":"Rudnick, Ze\u00e9v","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9781402054037"},{"key":"20","unstructured":"I. E. Shparlinski, Bounds for exponential sums with recurring sequences and their applications, Proc. Voronezh State Pedagogical Inst., 197 (1978), 74\u201385 (in Russian)."},{"key":"21","unstructured":"P. Sz\u00fcsz, On a problem in the theory of uniform distribution, Comptes Rendus Premier Congr\u00e8s Hongrois, Budapest, 1952, 461\u2013472 (in Hungarian)."},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"942","DOI":"10.1112\/plms.12204","article-title":"Nested efficient congruencing and relatives of Vinogradov\u2019s mean value theorem","volume":"118","author":"Wooley, Trevor D.","year":"2019","journal-title":"Proc. Lond. Math. Soc. 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