[Paper Review] Diophantine Approximations on Definable Sets

Philipp Habegger|arXiv (Cornell University)|Aug 16, 2016

Advanced Topology and Set Theory10 references3 citations

TL;DR

This paper establishes uniform bounds on the number of algebraic points of bounded degree and height that approximate a definable set in a polynomially bounded o-minimal structure to within a very small error. Using tools from o-minimality, Łojasiewicz inequalities, and auxiliary functions, it proves that such approximations cluster near the algebraic locus of the set, yielding a T^ε bound—complementing the Pila-Wilkie counting theorem for rational points on definable sets.

ABSTRACT

Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structure which give a general framework to work with sets mentioned above. It complements the theorem of Pila-Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We focus our attention on polynomially bounded o-minimal structures, allow algebraic points of bounded degree, and provide an estimate that is uniform over some families of definable sets. We apply these results to study fixed length sums of roots of unity that are small in modulus.

Motivation & Objective

• To extend the Pila-Wilkie counting theorem to rational and algebraic points that approximate definable sets rather than lie on them.
• To establish uniform bounds on the number of algebraic points of bounded degree and height that approximate a definable set within a small error margin.
• To work within polynomially bounded o-minimal structures to ensure effective and uniform estimates across families of definable sets.
• To apply the results to Diophantine problems, particularly bounding the number of small sums of roots of unity with rational angles.
• To demonstrate that such approximations are concentrated near the algebraic locus of the set, mirroring the Pila-Wilkie phenomenon for points on the set.

Proposed method

• Formalizes the notion of approximation via ε-neighborhoods of definable sets in o-minimal structures.
• Applies Łojasiewicz-type inequalities to control the distance from a point to the zero set of a real analytic function.
• Constructs an auxiliary function using the o-minimal structure to control the growth of derivatives and apply the Pila-Wilkie framework.
• Uses quasi-algebraic cells and induction schemes to decompose definable sets into manageable pieces with controlled complexity.
• Applies Northcott’s Theorem and Liouville’s Inequality to bound the number of algebraic points of bounded degree and height.
• Employs induction on the number of variables to analyze sums of roots of unity, reducing the problem to lower-dimensional cases with controlled error.

Experimental results

Research questions

• RQ1 How many algebraic points of bounded degree and height can approximate a definable set in a polynomially bounded o-minimal structure to within a very small error?
• RQ2 Can the Pila-Wilkie counting principle for rational points on definable sets be extended to rational approximations of such sets?
• RQ3 What is the uniformity of the T^ε bound across families of definable sets in o-minimal structures?
• RQ4 How do the approximations to a definable set relate to its algebraic locus in terms of density and distribution?
• RQ5 Can such bounds be applied to Diophantine problems such as bounding small sums of roots of unity?

Key findings

• For any closed definable set $X$ in a polynomially bounded o-minimal structure, the number of algebraic points of degree at most $e$ and height at most $T$ that approximate $X$ within $T^{- heta heta heta}$ is bounded by $cT^{ heta}$ for any $ heta > 0$.
• If $X$ has no positive-dimensional algebraic locus ($X^{ ext{alg}} = igcirc$), then the number of such approximating algebraic points is still bounded by $cT^{ heta}$, showing that approximations still cluster near algebraic structure.
• The bound is uniform across families of definable sets, with constants depending only on the set, degree $e$, and $ heta$.
• The results are applied to sums of roots of unity: the number of primes $p \leq T$ for which a fixed linear combination of $n$-th roots of unity has modulus less than $p^{-\lambda}$ is at most $cT^{\epsilon}$, for sufficiently large $c$ and $\lambda$.
• The proof relies on Łojasiewicz inequalities and induction, showing that if a sum is small, then the corresponding angle tuple must be close to a point in the algebraic locus of the zero set.
• The method avoids reliance on the full complexity of the original Pila-Wilkie theorem by focusing on approximation rather than containment, yielding a stronger uniformity in the error parameter.