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 structures 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.
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Acknowledgements
The author is indebted to important suggestions made by Jonathan Pila at an early stage of this work and to Felipe Voloch for pointing out a possible connection to small sums of roots of unity. He is grateful to Victor Beresnevich, David Masser, and Gerry Myerson for comments. He thanks Margaret Thomas and Alex Wilkie for their talks given in Manchester in 2015 and 2013, respectively. He also thanks the Institute for Advanced Study in Princeton, where this work was initiated at the end of 2013, for its hospitality. While there, he was supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Habegger, P. Diophantine approximations on definable sets. Sel. Math. New Ser. 24, 1633–1675 (2018). https://doi.org/10.1007/s00029-017-0378-7
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DOI: https://doi.org/10.1007/s00029-017-0378-7