On cardinality of character sums with Beatty sequences associated with composite modules
Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of characte...
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| Format: | Article |
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Tul'skii Gosudarstvennyi Pedagogicheskii Universitet imeni L.N. Tolstogo
2023
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| _version_ | 1825939887678816256 |
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| author | Ismail, Allakov Deraman, Fatanah Sapar, Siti Hasana Ismail, Shahrina |
| author_facet | Ismail, Allakov Deraman, Fatanah Sapar, Siti Hasana Ismail, Shahrina |
| author_sort | Ismail, Allakov |
| collection | UPM |
| description | Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of character sums and their estimation with respect to non homogeneous Beatty sequences beta alpha = [alpha n + beta : n = 1, 2, 3...] where beta in real numbers and alpha greater than zero is irrational. In order to estimate the cardinality, the discrepancy is used to measure the number of uniform distribution for Beatty sequences. Pigeonhole principle is discussed on the estimation of the fractional part of Beatty sequences involve. Meanwhile, Cauchy inequalities is applied to expand the double character sums. Then, the cardinality of double character sums is obtained by applying the extension properties of additive and multiplicative character sums. The result obtained is depend on the existing of identity of additive and multiplicative character sums and the uniformly distribution modulo 1. The result of the estimation in this study over composite modules is more general compared to previous studies, which only cover prime modules. |
| first_indexed | 2024-09-25T03:43:26Z |
| format | Article |
| id | upm.eprints-110596 |
| institution | Universiti Putra Malaysia |
| last_indexed | 2024-09-25T03:43:26Z |
| publishDate | 2023 |
| publisher | Tul'skii Gosudarstvennyi Pedagogicheskii Universitet imeni L.N. Tolstogo |
| record_format | dspace |
| spelling | upm.eprints-1105962024-06-17T08:35:11Z http://psasir.upm.edu.my/id/eprint/110596/ On cardinality of character sums with Beatty sequences associated with composite modules Ismail, Allakov Deraman, Fatanah Sapar, Siti Hasana Ismail, Shahrina Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of character sums and their estimation with respect to non homogeneous Beatty sequences beta alpha = [alpha n + beta : n = 1, 2, 3...] where beta in real numbers and alpha greater than zero is irrational. In order to estimate the cardinality, the discrepancy is used to measure the number of uniform distribution for Beatty sequences. Pigeonhole principle is discussed on the estimation of the fractional part of Beatty sequences involve. Meanwhile, Cauchy inequalities is applied to expand the double character sums. Then, the cardinality of double character sums is obtained by applying the extension properties of additive and multiplicative character sums. The result obtained is depend on the existing of identity of additive and multiplicative character sums and the uniformly distribution modulo 1. The result of the estimation in this study over composite modules is more general compared to previous studies, which only cover prime modules. Tul'skii Gosudarstvennyi Pedagogicheskii Universitet imeni L.N. Tolstogo 2023 Article PeerReviewed Ismail, Allakov and Deraman, Fatanah and Sapar, Siti Hasana and Ismail, Shahrina (2023) On cardinality of character sums with Beatty sequences associated with composite modules. Chebyshevskii Sbornik, 24 (5). pp. 5-15. ISSN 2226-8383 https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=cheb&paperid=1370&option_lang=eng 10.22405/2226-8383-2023-24-5-5-15 |
| spellingShingle | Ismail, Allakov Deraman, Fatanah Sapar, Siti Hasana Ismail, Shahrina On cardinality of character sums with Beatty sequences associated with composite modules |
| title | On cardinality of character sums with Beatty sequences associated with composite modules |
| title_full | On cardinality of character sums with Beatty sequences associated with composite modules |
| title_fullStr | On cardinality of character sums with Beatty sequences associated with composite modules |
| title_full_unstemmed | On cardinality of character sums with Beatty sequences associated with composite modules |
| title_short | On cardinality of character sums with Beatty sequences associated with composite modules |
| title_sort | on cardinality of character sums with beatty sequences associated with composite modules |
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