×

zbMATH — the first resource for mathematics

The number of integral points on arcs and ovals. (English) Zbl 0718.11048
Let \(C\) be an arc of a convex curve \(y=F(x)\) lying in the unit square, and let \(R(N)\) be the number of integer points on \(NC\) (rational points \((m/N,n/N)\) on \(C\)). The authors prove results of the form \[ R(N) < B(\varepsilon) N^{\theta +\varepsilon} \tag{1} \] for \(N\) sufficiently large. V. Jarník [Math. Z. 24, 500–518 (1925; JFM 51.0153.01)] constructed curves with \(R(N)\geq AN^{2/3}\) for any given \(N\), so the conditions “\(N\) sufficiently large” and “\(B(\varepsilon)\) depending on \(C\)” cannot both be dropped. H. P. F. Swinnerton-Dyer [J. Number Theory 6, 128–135 (1974; Zbl 0285.10020)] gave (1) with \(\theta =3/5\) when \(F(x)\) has three continuous derivatives. W. M. Schmidt [Monatsh. Math. 99, 45–72 (1985; Zbl 0551.10026)] made \(B(\varepsilon)\) independent of \(C\) for \(F^{(3)}\) non-vanishing.
Algebraic curves parametrised by polynomials of degree \(d\) over \({\mathbb Q}\) have \(R(N)\geq AN^{1/d}\) for infinitely many \(N\). The authors show (Theorems 1 and 2) that all other real-analytic curves satisfy (1) with \(\theta =0\). The constant \(B(\varepsilon)\) depends on \(C\) ineffectively. For irreducible algebraic curves of degree \(d\), (1) holds with \(\theta =1/d\) and \(B(\varepsilon)\) depending only on \(d\) (Theorem 5). For an arbitrary curve (1) holds with \(\theta =1/2+8/3(d+3),\) provided that \(F(x)\) has \(D=(d+1)(d+2)/2\) continuous derivatives, with \(B(\varepsilon)\) depending on upper bounds for the derivatives of \(F(x)\) (Theorem 6). Theorem 8 is a slightly weaker result in which \(B(\varepsilon)\) depends on the number of zeros of \(F^{(D)}(x)\). For \(D\geq 325\) Swinnerton-Dyer’s exponent is greatly improved. The authors show that \(\varepsilon\) is needed in the exponent by constructing an infinitely differentiable \(F(x)\) for any given value of \(N\), with \(F^{(3)}(x)\) non-negative, but \(R(N)\geq A(N \log N)^{1/2}.\)
The method is to show that if \(D\) integer points lie close on a smooth curve, then a certain determinant of monomials in the coordinates is small. Being an integer, the determinant must be zero, and the \(D\) points lie on an algebraic curve of degree \(d\) (for algebraic \(C\) the method is modified to ensure that \(C\) is not a component). The effective results use induction on the degree and intersection number arguments.

MSC:
11P21 Lattice points in specified regions
11D99 Diophantine equations
11H06 Lattices and convex bodies (number-theoretic aspects)
11G99 Arithmetic algebraic geometry (Diophantine geometry)
14G05 Rational points
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] E. Bombieri, Le grand crible dans la théorie analytique des nombres , Astérisque (1987), no. 18, 103, Soc. Math. France, Paris. · Zbl 0618.10042
[2] S. D. Cohen, The distribution of Galois groups and Hilbert’s irreducibility theorem , Proc. London Math. Soc. (3) 43 (1981), no. 2, 227-250. · Zbl 0484.12002
[3] D. Hilbert and A. Hurwitz, Über die diophantischen Gleichungen vom Geschlecht Null , Acta Mathematica 14 (1890-1891), 217-224.
[4] V. Jarnik, Über die Gitterpunkte auf konvexen Curven , Math. Z. 24 (1926), 500-518. · JFM 51.0153.01
[5] D. J. Lewis and K. Mahler, On the representation of integers by binary forms , Acta Arith. 6 (1961), 333-363. · Zbl 0102.03601
[6] C. Posse, Sur le terme complémentaire de la formule de M. Tchebychef donnant l’expression approchée d’une intégrale définie par d’autres prises entre les mêmes limites , Bull. Sci. Math. (2) 7 (1883), 214-224. · JFM 15.0237.01
[7] P. Sarnak, Torsion points on varieties and homology of Abelian covers , manuscript, 1988.
[8] W. M. Schmidt, Integer Points on Curves and Surfaces , Monatsh. Math. 99 (1985), no. 1, 45-72. · Zbl 0551.10026
[9] H. A. Schwarz, Verallgemeinerung eines analytischen Fundamentalsatzes , Annali di Mat. (2) 10 (1880), 129-136, rpt. Gesammelte Mathematische Abhandlungen, vol. 2, J. Springer, Berlin, 1890, pp. 296-302.
[10] H. P. F. Swinnerton-Dyer, The number of lattice points on a convex curve , J. Number Theory 6 (1974), 128-135. · Zbl 0285.10020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.