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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.

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
Full Text: DOI Euclid
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