{"title":"基-q和Ostrowski数字和的剩余类的联合分布","authors":"Divyum Sharma","doi":"10.2478/udt-2019-0010","DOIUrl":null,"url":null,"abstract":"Abstract Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.For α=[ 0;1,m¯ ], m≥2, \\alpha = \\left[ {0;\\overline {1,m} } \\right],\\,\\,\\,m \\ge 2, let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers with gcd(q-1,m1)=gcd(m,m2)=1 \\gcd \\left( {q - 1,{m_1}} \\right) = \\gcd \\left( {m,{m_2}} \\right) = 1 We prove that there exists δ> 0 such that for all integers r1,r2, | { 0≤n<N:Sq(n)≡r1(mod m1), Sα(n)≡r2(mod m2) }|=Nm1m2+0(N1-δ). \\matrix{ {\\left| {\\left\\{ {0 \\le n < N:{S_q}(n) \\equiv {r_1}\\left( {\\bmod \\,{m_1}} \\right),\\,\\,{S_\\alpha }(n) \\equiv {r_2}\\left( {\\bmod \\,{m_2}} \\right)} \\right\\}} \\right|} \\cr { = {N \\over {{m_1}{m_2}}} + 0\\left( {{N^{1 - \\delta }}} \\right).} \\cr } The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[ 1¯ ] \\alpha = \\left[ {\\bar 1} \\right] by Spiegelhofer.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"81 1","pages":"1 - 26"},"PeriodicalIF":0.0000,"publicationDate":"2017-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums\",\"authors\":\"Divyum Sharma\",\"doi\":\"10.2478/udt-2019-0010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.For α=[ 0;1,m¯ ], m≥2, \\\\alpha = \\\\left[ {0;\\\\overline {1,m} } \\\\right],\\\\,\\\\,\\\\,m \\\\ge 2, let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers with gcd(q-1,m1)=gcd(m,m2)=1 \\\\gcd \\\\left( {q - 1,{m_1}} \\\\right) = \\\\gcd \\\\left( {m,{m_2}} \\\\right) = 1 We prove that there exists δ> 0 such that for all integers r1,r2, | { 0≤n<N:Sq(n)≡r1(mod m1), Sα(n)≡r2(mod m2) }|=Nm1m2+0(N1-δ). \\\\matrix{ {\\\\left| {\\\\left\\\\{ {0 \\\\le n < N:{S_q}(n) \\\\equiv {r_1}\\\\left( {\\\\bmod \\\\,{m_1}} \\\\right),\\\\,\\\\,{S_\\\\alpha }(n) \\\\equiv {r_2}\\\\left( {\\\\bmod \\\\,{m_2}} \\\\right)} \\\\right\\\\}} \\\\right|} \\\\cr { = {N \\\\over {{m_1}{m_2}}} + 0\\\\left( {{N^{1 - \\\\delta }}} \\\\right).} \\\\cr } The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[ 1¯ ] \\\\alpha = \\\\left[ {\\\\bar 1} \\\\right] by Spiegelhofer.\",\"PeriodicalId\":23390,\"journal\":{\"name\":\"Uniform distribution theory\",\"volume\":\"81 1\",\"pages\":\"1 - 26\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-10-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Uniform distribution theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/udt-2019-0010\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Uniform distribution theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/udt-2019-0010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums
Abstract Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.For α=[ 0;1,m¯ ], m≥2, \alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2, let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers with gcd(q-1,m1)=gcd(m,m2)=1 \gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1 We prove that there exists δ> 0 such that for all integers r1,r2, | { 0≤n
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
