Integer points on the dilation of a subanalytic surface.

*(English)*Zbl 1111.32004Let \(\Omega\) 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.

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.

Reviewer: Carles Biviá-Ausina (València)