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The density of rational points near hypersurfaces. (English) Zbl 1458.11115

Author’s description of the present research is following:
“We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality, and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre’s dimension growth conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we solve the generalized Baker-Schmidt problem in the simultaneous setting for hypersurfaces.”
The author notes that investigations of integral and rational points on algebraic varieties are a fundamental theme in mathematics. This article deals with Diophantine geometry and with the distribution of rational points near manifolds. The main attention is given to the following main problem and its applications to a Diophantine inequality, to the dimension growth conjecture, and to metric Diophantine approximation on manifolds.
The main problem. “Estimate \(N_{\mathcal{M}}(Q,\delta)\) for a ‘generic’ manifold \(\mathcal{M}\).”
Here, \(\mathcal{M}\) is a bounded submanifold of \(\mathbb{R}^n\) of dimension \(m\). Also, \(Q\ge2\) and \(\delta\in(0,1/2)\) are given, and \[ N_{\mathcal{M}}(Q,\delta):=\#\left\{{\mathbf{p}}/q\in\mathbb{Q}^n: 1\le q\le Q, \mathrm{dist}_\infty\left({\mathbf{p}}/q,\mathcal{M}\right)\le{\delta}/q\right\}, \] where \(\mathbf{p}\in\mathbb{Z}^n, q\in\mathbb{Z}\) and \(\mathrm{dist}_\infty(\cdot,\cdot)\) is the \(L^\infty\) distance on \(\mathbb{R}^n\).
A brief survey on the history of this problem in the literature is given. The main results and several auxiliary statements are proven with explanations and some examples are considered. Auxiliary notions are recalled. The main attention is given to results related to the following topics: a Diophantine inequality of Robert and Sargos, Serre’s dimension growth conjecture, and Diophantine approximations on manifolds.

MSC:

11J83 Metric theory
11J13 Simultaneous homogeneous approximation, linear forms
11J25 Diophantine inequalities
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References:

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