From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits

By Alex Kontorovich. 2012

68 citations recorded by the source corpus.

We discuss a number of naturally arising problems in arithmetic, culled from completely unrelated sources, which turn out to have a common formulation involving "thin" orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/dd3b1bff4c6e885ec09d390ead1a3aa6909df90d

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In Algebra and Number Theory

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

Zaremba’s conjecture (2011, methodology): “: Zaremba’s conjecture has important applications to numerical integration and pseudorandom number generation, producing collections of points of optimal discrepancy; see e.g. the surveys [Nie78] and [Kon13]. Our main result is the following Theorem 1.2. Almost every positive integer is the denomina”
Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of E2 (2016, unknown): “[20,23]); in particular they are Cantor sets, and study of their Hausdorff dimension has attracted significant attention.”
Schubert calculus and torsion explosion (2013, background): “on of the Aﬃne Sieve [BGS06, BGS10, SGS13]; see also, e.g., [Kon14]. It turns out that one can do much more using recent progress on“local-global”problems in“thin orbits”(see, e.g., the discussion in [Kon13]); namely, one can produce not just prime divisors but actual primes in the entries of Γ, and”
An Improvement to Zaremba’s Conjecture (2013, background): “Zaremba’s conjecture has been closely related to numerical integration and pseudorandom number generation, see [Nie78] and [Kon13].”