ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY
Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$ ) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$...
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| Veröffentlicht in: | Mathematika 2018, Vol.64 (3), p.679-700 |
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| container_title | Mathematika |
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| creator | Bloom, Thomas F. Chow, Sam Gafni, Ayla Walker, Aled |
| description | Let
$A$
be a set of natural numbers. Recent work has suggested a strong link between the additive energy of
$A$
(the number of solutions to
$a_{1}+a_{2}=a_{3}+a_{4}$
with
$a_{i}\in A$
) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of
$A$
modulo
$1$
. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian. |
| doi_str_mv | 10.1112/S0025579318000207 |
| format | Article |
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$A$
be a set of natural numbers. Recent work has suggested a strong link between the additive energy of
$A$
(the number of solutions to
$a_{1}+a_{2}=a_{3}+a_{4}$
with
$a_{i}\in A$
) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of
$A$
modulo
$1$
. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.</description><identifier>ISSN: 0025-5793</identifier><identifier>EISSN: 2041-7942</identifier><identifier>DOI: 10.1112/S0025579318000207</identifier><language>eng</language><publisher>London, UK: London Mathematical Society</publisher><subject>05B10 ; 11B30 ; 11J71 ; 11J83 ; 60F10 (primary)</subject><ispartof>Mathematika, 2018, Vol.64 (3), p.679-700</ispartof><rights>Copyright © University College London 2018</rights><rights>2018 University College London</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c3927-702d73d1c083e677caf58985ff2344b6446e5c4870a20b8fce9a7995f197cffe3</citedby><cites>FETCH-LOGICAL-c3927-702d73d1c083e677caf58985ff2344b6446e5c4870a20b8fce9a7995f197cffe3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1112%2FS0025579318000207$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://www.cambridge.org/core/product/identifier/S0025579318000207/type/journal_article$$EHTML$$P50$$Gcambridge$$H</linktohtml><link.rule.ids>164,314,776,780,1411,4010,27900,27901,27902,45550,45551,55603</link.rule.ids></links><search><creatorcontrib>Bloom, Thomas F.</creatorcontrib><creatorcontrib>Chow, Sam</creatorcontrib><creatorcontrib>Gafni, Ayla</creatorcontrib><creatorcontrib>Walker, Aled</creatorcontrib><title>ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY</title><title>Mathematika</title><addtitle>Mathematika</addtitle><description>Let
$A$
be a set of natural numbers. Recent work has suggested a strong link between the additive energy of
$A$
(the number of solutions to
$a_{1}+a_{2}=a_{3}+a_{4}$
with
$a_{i}\in A$
) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of
$A$
modulo
$1$
. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.</description><subject>05B10</subject><subject>11B30</subject><subject>11J71</subject><subject>11J83</subject><subject>60F10 (primary)</subject><issn>0025-5793</issn><issn>2041-7942</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNqFj11Kw0AUhQdRMFYX4Fs2EL3zl5l5DMnYBtukpFHoU0gnM5LSWkkQ6W5ciyszpX1T9OleOHzn8CF0i-EOY0zuFwCEc6EoljC8IM6QR4DhQChGzpF3iINDfomu-n4NwEPJsIdolCRpmT5rX2e6GC_9KEv8cqL9mS6LNPbnebpY5FkaZV-f8yKf66JcXqMLV296e3O6I_T0oMt4EkzzcRpH08BQRUQggDSCNtiApDYUwtSOSyW5c4QytgoZCy03TAqoCaykM1bVQinusBLGOUtHCB97Tbfr-8666q1rt3W3rzBUB-vqh_XAxEfmo93Y_f9ANSsff2uhp-V6u-ra5sVW69179zrY_rH9DSHvZmQ</recordid><startdate>2018</startdate><enddate>2018</enddate><creator>Bloom, Thomas F.</creator><creator>Chow, Sam</creator><creator>Gafni, Ayla</creator><creator>Walker, Aled</creator><general>London Mathematical Society</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2018</creationdate><title>ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY</title><author>Bloom, Thomas F. ; Chow, Sam ; Gafni, Ayla ; Walker, Aled</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3927-702d73d1c083e677caf58985ff2344b6446e5c4870a20b8fce9a7995f197cffe3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>05B10</topic><topic>11B30</topic><topic>11J71</topic><topic>11J83</topic><topic>60F10 (primary)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bloom, Thomas F.</creatorcontrib><creatorcontrib>Chow, Sam</creatorcontrib><creatorcontrib>Gafni, Ayla</creatorcontrib><creatorcontrib>Walker, Aled</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematika</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bloom, Thomas F.</au><au>Chow, Sam</au><au>Gafni, Ayla</au><au>Walker, Aled</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY</atitle><jtitle>Mathematika</jtitle><addtitle>Mathematika</addtitle><date>2018</date><risdate>2018</risdate><volume>64</volume><issue>3</issue><spage>679</spage><epage>700</epage><pages>679-700</pages><issn>0025-5793</issn><eissn>2041-7942</eissn><abstract>Let
$A$
be a set of natural numbers. Recent work has suggested a strong link between the additive energy of
$A$
(the number of solutions to
$a_{1}+a_{2}=a_{3}+a_{4}$
with
$a_{i}\in A$
) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of
$A$
modulo
$1$
. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.</abstract><cop>London, UK</cop><pub>London Mathematical Society</pub><doi>10.1112/S0025579318000207</doi><tpages>22</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5793 |
| ispartof | Mathematika, 2018, Vol.64 (3), p.679-700 |
| issn | 0025-5793 2041-7942 |
| language | eng |
| recordid | cdi_crossref_primary_10_1112_S0025579318000207 |
| source | Wiley Online Library Journals Frontfile Complete; Cambridge Journals |
| subjects | 05B10 11B30 11J71 11J83 60F10 (primary) |
| title | ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY |
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