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Integer points on the dilation of a subanalytic surface. (English) Zbl 1111.32004
Let $$\Omega$$ be a subset of $$\mathbb R^n$$, and let us denote the intersection $$\Omega\cap \mathbb Z^n$$ by $$\Omega(\mathbb Z)$$ and the intersection $$\Omega\cap\mathbb Q^n$$ by $$\Omega (\mathbb Q)$$. This paper is concerned with the study of the control exerted by the geometry of $$\Omega$$ on the structure of $$\Omega (\mathbb Z)$$ and $$\Omega (\mathbb Q)$$ (see also the monographs of S. Lang [Number theory III: Diophantine geometry (1991; Zbl 0744.14012)] and M. Hindry and J. H. Silverman [Diophantine geometry. An introduction (2000; Zbl 0948.11023)]).
Suppose that $$\Omega$$ is a subanalytic set. If the dimension of $$\Omega$$ is at least $$2$$, then $$\Omega$$ may contain subsets of positive dimension that are semialgebraic even if $$\Omega$$ itself is not semialgebraic (in Section 2 of the paper the definition of subanalytic set and known facts about subanalytic sets are given). Let $$\Omega^{\text{alg}}$$ be the union of all connected subanalytic sets of $$\Omega$$ of positive dimension that are semialgebraic, and let $$\Omega^{\text{trans}}=\Omega\smallsetminus\Omega^{\text{alg}}$$. If $$t\geq 1$$, the homothetic dilation $$t\Omega$$ of $$\Omega$$ by $$t$$ is defined by $$t\Omega=\{tx: x\in\Omega\}$$. Moreover, given a point $$P\in\mathbb Q^n$$, suppose that $$P$$ is written as $$P=(a_1/b, \dots, a_n/b)$$, for some $$a_1,\dots, a_n, b\in\mathbb Z$$, where $$b>0$$ and $$\text{gcd}(a_1,\dots, a_n,b)=1$$. The height of $$P$$ is defined as $$H(P)=\max\{| a_j|,b\}$$. Therefore, for $$H\geq 1$$, the set of those $$P\in \Omega(\mathbb Q)$$ such that $$H(P)\leq H$$ is denoted by $$\Omega(\mathbb Q,H)$$. The author states the following conjectures:
Conjecture 1: For compact subanalytic $$\Omega\subset\mathbb R^n$$ and $$\epsilon>0$$ there is a constant $$c_1(\Omega,\epsilon)$$ such that $\#t\Omega^{\text{trans}}(\mathbb Z)\leq c_1(\Omega,\epsilon)t^\epsilon.$
Conjecture 2: For compact subanalytic $$\Omega\subset\mathbb R^n$$ and $$\epsilon>0$$ there is a constant $$c_2(\Omega,\epsilon)$$ such that $\#\Omega^{\text{trans}}(\mathbb Q)\leq c_2(\Omega,\epsilon)H^\epsilon.$
One of the main results of the paper is the proof of Conjecture 1 for compact subanalytic surfaces. It is also proven that, if $$\epsilon >0$$ and $$\Omega$$ is a compact analytic submanifold of $$\mathbb R^n$$ of dimension 2 such that $$\Omega^{\text{alg}}$$ is semialgebraic of dimension 1, then there is a constant $$c(\Omega, \epsilon)$$ such that $\#t\Omega(\mathbb Z)\leq c(\Omega, \epsilon)t^\epsilon.$
In Section 7 of the paper the author proves Conjectures 1 and 2 for compact subanalytic curves. The main results are proven in Section 8. For those proofs the author uses a key result implicit in the paper of E. Bombieri and Pila [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)], this result is recalled in Section 8.

##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 11D99 Diophantine equations 11J99 Diophantine approximation, transcendental number theory
##### Keywords:
subanalytic sets; semialgebraic sets; Diophantine equations
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