Abstract
In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For $n \geq 3$, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in $\mathbb{R}^n$.
Highlights
Introduction and Statement of Results1.1 The setup Throughout, we suppose that m ≤ d, n = m + d and that f = (f1, . . . , fm) is defined on U = [0, 1]d
Throughout, the Vinogradov symbols ≪ and ≫ will be used to indicate an inequality with an unspecified positive multiplicative constant
Throughout the article, the constants will only depend on the dimensions n and d and the map f
Summary
Throughout, the Vinogradov symbols ≪ and ≫ will be used to indicate an inequality with an unspecified positive multiplicative constant. Throughout the article, the constants will only depend on the dimensions n and d and the map f. We establish the following upper bound result. Suppose that f : U → Rm satisfies (1.1) and θ ∈ Rn. Suppose that 0 < ψ(q) ≤ 1/2. Where the implied constant is independent of q, θ, and ψ but may depend on f. The following is a straightforward consequence of the theorem. It states that the upper bound (1.6) coincides with the heuristic estimate if ψ(q) is not too small. Suppose that f : U → Rm satisfies (1.1) and θ ∈ Rn. Suppose that q−1/(2m+1)(log q)2/(2m+1) ≤ ψ (q) ≤ 1/2.
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