Rational points on the sphere
نویسندگان
چکیده
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 least common denominator of its coordinates in reduced form. First we shall count the rational points of height ≤ T on S and show that they become uniformly distributed with respect to (normalized) Lebesgue measure μ on S as T →∞. Then we shall show that the rational points of a given height become uniformly distributed as the height tends to infinity through odd values.
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