On the size of finite Sidon sets
UDC 519.1 A Sidon set (also called a Golomb ruler) is a $B_2$ sequence and a $1$-thin set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d.$ We improve the lower bound for the diameter of a Sidon set with $k$ elements, namely, if $k$ is sufficiently large and $\mathca...
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| Datum: | 2024 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7858 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512787106627584 |
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| author | O'Bryant, Kevin O'Bryant, Kevin |
| author_facet | O'Bryant, Kevin O'Bryant, Kevin |
| author_sort | O'Bryant, Kevin |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-09-25T06:57:35Z |
| description | UDC 519.1
A Sidon set (also called a Golomb ruler) is a $B_2$ sequence and a $1$-thin set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d.$ We improve the lower bound for the diameter of a Sidon set with $k$ elements, namely, if $k$ is sufficiently large and $\mathcal A$ is a Sidon set with $k$ elements, then ${\rm diam}({\mathcal A})\ge k^2-1.99405 k^{3/2}.$ Alternatively, if $n$ is sufficiently large, then the cardinality of the largest subset of $\{1,2,\dots,n\},$ which is a Sidon set, does not exceed $n^{1/2}+0.99703 n^{1/4}.$ |
| doi_str_mv | 10.3842/umzh.v76i8.7858 |
| first_indexed | 2026-03-24T03:34:20Z |
| format | Article |
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| id | umjimathkievua-article-7858 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:34:20Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-78582024-09-25T06:57:35Z On the size of finite Sidon sets On the size of finite Sidon sets O'Bryant, Kevin O'Bryant, Kevin Sidon Set Golomb Ruler Bh set Number Theory Combinatorics UDC 519.1 A Sidon set (also called a Golomb ruler) is a $B_2$ sequence and a $1$-thin set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d.$ We improve the lower bound for the diameter of a Sidon set with $k$ elements, namely, if $k$ is sufficiently large and $\mathcal A$ is a Sidon set with $k$ elements, then ${\rm diam}({\mathcal A})\ge k^2-1.99405 k^{3/2}.$ Alternatively, if $n$ is sufficiently large, then the cardinality of the largest subset of $\{1,2,\dots,n\},$ which is a Sidon set, does not exceed $n^{1/2}+0.99703 n^{1/4}.$ УДК 519.1 Про розмір скінченних множин Сідона Множина Сідона (що також називається лінійкою Голомба) –  це послідовність $B_2$, а $1$-тонка множина – це множина цілих чисел, що не містить нетривіальних розв'язків рівняння $a+b=c+d.$ Ми покращуємо нижню межу для діаметра множини Сідона з $k$ елементів, тобто якщо $k$ є достатньо великим, а $\cal A$ є множиною Сідона з $k$ елементами, то ${\rm diam}({\cal A}) \ge k^2-1,99405 k^{3/2}.$ Навпаки, якщо $n$ достатньо велике, то потужність найбільшої підмножини $\{1,2,\dots,n\}$, що є множиною Сідона,  не перевищує $n^{1/2}+0.99703 n^ {1/4}.$ Institute of Mathematics, NAS of Ukraine 2024-09-04 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7858 10.3842/umzh.v76i8.7858 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 8 (2024); 1192 - 1206 Український математичний журнал; Том 76 № 8 (2024); 1192 - 1206 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7858/10155 Copyright (c) 2024 Kevin O'Bryant |
| spellingShingle | O'Bryant, Kevin O'Bryant, Kevin On the size of finite Sidon sets |
| title | On the size of finite Sidon sets |
| title_alt | On the size of finite Sidon sets |
| title_full | On the size of finite Sidon sets |
| title_fullStr | On the size of finite Sidon sets |
| title_full_unstemmed | On the size of finite Sidon sets |
| title_short | On the size of finite Sidon sets |
| title_sort | on the size of finite sidon sets |
| topic_facet | Sidon Set Golomb Ruler Bh set Number Theory Combinatorics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7858 |
| work_keys_str_mv | AT o039bryantkevin onthesizeoffinitesidonsets AT o039bryantkevin onthesizeoffinitesidonsets |