Integral points on a very flat convex curve

International audienceThe 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