The Idea for this article was given by a problem in real analysis. We wanted to determine the one-dimensional Lebesgue-measure of the set f(C)9 where C stands for the classical triadic Cantor set and/is the Cantor-function, which is also known as "devil's staircase." We could see immediately that to determine the above measure we needed to know which dyadic rationals were contained in C. We soo...

For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in P. This paper is concerned with the number N(F ; B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F ; B) = O(B), whenever either n ≤ 5 or the hypersurface is not a union of lines. Here the implied constant depends at ...

S. Lang [L] conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Nog...

Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over Q. We show that X(Q) is non-empty provided that X(R) is non-empty and X has p-adic points for every prime p.

The object of this paper is to give a short and reasonably self-contained treatment of the distribution properties of the rational points on the sphere S = {x ∈ R;x1 + x2 + x3 = 1} using only basic properties of modular forms, including the Shimura lift and the fundamental work of Rankin. Consider the set R of rational points Q ∩ S. By the height h(x) of a point x ∈ R we shall mean simply the l...

In this paper, we study a geometric G^2 Hermite interpolation by planar rational cubic Bézier curves. Two data points, two tangent vectors and two signed curvatures interpolated per each rational segment. We give the necessary and the sufficient intrinsic geometric conditions for two C^2 parametric curves to be connected with G2 continuity. Locally, the free parameters w...