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Wilkie’s conjecture for restricted elementary functions. (English) Zbl 1383.11090
In the paper under review, the authors prove Wilkie’s conjecture for restricted elementary functions. The problem settled by the authors goes back to works by E. Bombieri and J. Pila [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)] concerned with answering a question by Sarnak on Betti numbers of abelian covers. Among other things, they proved that, given a transcendental analytic function \(f : [0, 1] \rightarrow {\mathbb R}\) with the graph \(X \subset {\mathbb R}^2\) its graph, for every \(\varepsilon > 0\) there exists a constant \(c(f,\varepsilon)\) such that the density function satisfies \(\sharp (tX \cap {\mathbb Z}^2) \leq c(f,\varepsilon) t^{\varepsilon}\) for all \(t \geq 1\). The proof introduced a new method of counting integer points using certain interpolation determinants. In [Duke Math. J. 63, No. 2, 449–463 (1991; Zbl 0763.11025)], J. Pila extended this method to the problem of counting rational points on \(X\) and proved that if \(f\) is transcendental then, for every \(\varepsilon > 0\), there exists a constant \(c(f, \varepsilon)\) such that \(\sharp X({\mathbb Q}, H) \leq c(f, \varepsilon) H^{\varepsilon}\) for all \(H \in {\mathbb N}\), and in [Ann. Inst. Fourier 55, No. 5, 1501–1516 (2005; Zbl 1121.11032)] he showed that, for compact subanalytic surfaces \(X \subset {\mathbb R}^n\) and \(\varepsilon > 0\), there exists a constant \(C(X,\varepsilon)\) such that \(\sharp X^{\mathrm{trans}}({\mathbb Q}, H) \leq C(X, \varepsilon) H^{\varepsilon}\), as well as conjectured the result to hold for all compact subanalytic sets. This was, in fact, affirmed by J. Pila and A. J. Wilkie [Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)], where a much more general version of the problem was settled: the authors proved that if the set \(X\) is definable in any o-minimal structure and if \(\varepsilon > 0\), then there exists a constant \(C(X, \varepsilon)\) such that \(\sharp X^{\mathrm{trans}}({\mathbb Q}, H) \leq C(X, \varepsilon)H^{\varepsilon}\). In the same paper, the authors posed a question now known as the Wilkie’s conjecture whether for sets \(X\) definable in \({\mathbb R}_{\mathrm{exp}}\) (that is, using the unrestricted exponential but without allowing arbitrary restricted analytic functions), there exist constants \(N(X)\) and \(\kappa(X)\) such that: \[ \sharp X^{\mathrm{trans}} ({\mathbb Q},H) \leq N(X) \cdot (\log H)^{\kappa(X)}. \] Generalizations of the conjecture for number fields were also proposed by J. Pila in [Ann. Inst. Fourier 60, No. 2, 489–514 (2010; Zbl 1210.11074)]. Some low-dimensional cases of the Wilkie conjecture have been established: for graphs of Pfaffian functions by J. Pila [Ann. Fac. Sci. Toulouse, Math. (6) 16, No. 3, 635–645 (2007; Zbl 1229.11053)], for surfaces definable in the structure of restricted Pfaffian functions by G. O. Jones and M. E. M. Thomas [Q. J. Math. 63, No. 3, 637–651 (2012; Zbl 1253.03065)], or for some special surfaces defined using the unrestricted exponential by J. Pila [Ann. Inst. Fourier 60, No. 2, 489–514 (2010; Zbl 1210.11074)]. In the present paper, the authors settle the Wilkie’s conjecture for sets \(X\) definable in the structure \({\mathbb R}^{RE}\) obtained from \(({\mathbb R}, <, +, \cdot)\) by adjoining the restricted exponential and sine functions, as well as two versions of the conjecture concerned with the density of algebraic points from a fixed number field. The methods applied by the authors make extensive use of the category of holomorphic Pfaffian functions, that is holomorphic functions whose graphs are projections of sets defined by Boolean combinations of Pfaffian equalities and inequalities.

MSC:
11G99 Arithmetic algebraic geometry (Diophantine geometry)
03C64 Model theory of ordered structures; o-minimality
11U09 Model theory (number-theoretic aspects)
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