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Integral points on a very flat convex curve
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | DESHOUILLERS, Jean-Marc | |
| hal.structure.identifier | Combinatoire, théorie des nombres [CTN] | |
| dc.contributor.author | GREKOS, Georges | |
| dc.date.accessioned | 2024-04-04T03:01:30Z | |
| dc.date.available | 2024-04-04T03:01:30Z | |
| dc.date.issued | 2017 | |
| dc.identifier.isbn | 978-3-319-68376-8 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/192908 | |
| dc.description.abstractEn | The second named author studied in 1988 the possible relations between the length , the minimal radius of curvature r and the number of integral points N of a strictly convex flat curve in R 2 , stating that N = O(/r 1/3) (*), a best possible bound even when imposing the tangent at one extremity of the curve; here flat means that one has = r α for some α ∈ [2/3, 1). He also proved that when α ≤ 1/3, the quantity N is bounded. In this paper, the authors prove that in general the bound (*) cannot be improved for very flat curves, i.e. those for which α ∈ (1/3, 2/3); however, if one imposes a 0 tangent at one extremity of the curve, then (*) is replaced by the sharper inequality N ≤ 2 /r+1. Abstract. The second named author studied in 1988 the possible relations between the length , the minimal radius of curvature r and the number of integral points N of a strictly convex flat curve in R 2 , stating that N = O(/r 1/3) (*), a best possible bound even when imposing the tangent at one extremity of the curve; here flat means that one has = r α for some α ∈ [2/3, 1). He also proved that when α ≤ 1/3, the quantity N is bounded. In this paper, the authors prove that in general the bound (*) cannot be improved for very flat curves, i.e. those for which α ∈ (1/3, 2/3); however, if one imposes a 0 tangent at one extremity of the curve, then (*) is replaced by the sharper inequality N ≤ 2 /r + 1. | |
| dc.language.iso | en | |
| dc.source.title | Analytic Number Theory, Modular Forms and q-Hypergeometric Series | |
| dc.subject.en | geometry of numbers | |
| dc.subject.en | integer points | |
| dc.subject.en | strictly convex curves | |
| dc.title.en | Integral points on a very flat convex curve | |
| dc.type | Chapitre d'ouvrage | |
| dc.subject.hal | Mathématiques [math]/Théorie des nombres [math.NT] | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.title.proceeding | Analytic Number Theory, Modular Forms and q-Hypergeometric Series | |
| hal.identifier | hal-02084722 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02084722v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.btitle=Analytic%20Number%20Theory,%20Modular%20Forms%20and%20q-Hypergeometric%20Series&rft.date=2017&rft.au=DESHOUILLERS,%20Jean-Marc&GREKOS,%20Georges&rft.isbn=978-3-319-68376-8&rft.genre=unknown |
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