DESHOUILLERS, Jean-Marc; GREKOS, Georges

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DESHOUILLERS, Jean-Marc
Institut de Mathématiques de Bordeaux [IMB]

GREKOS, Georges
Combinatoire, théorie des nombres [CTN]

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Chapitre d'ouvrage

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Analytic Number Theory, Modular Forms and q-Hypergeometric Series, Analytic Number Theory, Modular Forms and q-Hypergeometric Series. 2017

Résumé en anglais

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.< Réduire

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geometry of numbers

integer points

strictly convex curves

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Integral points on a very flat convex curve

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