On the local-global conjecture for integral Apollonian gaskets

Abstract

Abstract. We prove that a set of density one satisfies the localglobal conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.

Local-global principles in circle packings

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We generalize work of Bourgain-Kontorovich [6] and Zhang [32], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group A ≤ PSL 2 (K) satisfying certain conditions, where K is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle.

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A Remark on Sieving in Biased Coin Convolutions

“…(See e.g. [1], [2], [3], [4]) we have the following result about r-pseudo-primes (products of at most r primes).…”

“…Examples of natural appearances of this type of questions include the study of the curvatures in integral Apollonian circle packings, Pythagorean triples and issues around fundamental discriminates of quadratic number fields and low lying geodesics in the modular surface. (See [2].) The reader may also wish to consult the excellent Bourbaki exposition by E. Kowalski [6] for a detailed account of many of these recent developments around 'exotic sieving'.…”

Statistical Regularity of Apollonian Gaskets

“…The last 15 years have overseen tremendous progress in understanding the structure of Apollonian gaskets from different viewpoints, such as number theory and geometry [16], [15], [10], [11], [20], [25]. In the geometric direction, generalizing a result of [20], Hee Oh and Nimish Shah proved the following remarkable theorem concerning the growth of circles.…”