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Almost Primes in Thin Orbits of Pyth...

Ehrman, Max.

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Almost Primes in Thin Orbits of Pythagorean Triangles.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Almost Primes in Thin Orbits of Pythagorean Triangles./
Author:	Ehrman, Max.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2017,
Description:	59 p.
Notes:	Source: Dissertation Abstracts International, Volume: 78-11(E), Section: B.
Contained By:	Dissertation Abstracts International78-11B(E).
Subject:	Theoretical mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10631581
ISBN:	9780355017762

Almost Primes in Thin Orbits of Pythagorean Triangles.
Ehrman, Max.

Almost Primes in Thin Orbits of Pythagorean Triangles. - Ann Arbor : ProQuest Dissertations & Theses, 2017 - 59 p.

Source: Dissertation Abstracts International, Volume: 78-11(E), Section: B.

Thesis (Ph.D.)--Yale University, 2017.

This item is not available from ProQuest Dissertations & Theses.

Let F = x2 + y2 -- z2, and let x0 epsilon Z 3 be a primitive solution to F(x0) = 0, e.g. so that its coordinates share no nontrivial divisor. Let Gamma ≤ SOF( Z ) be a finitely generated thin subgroup, one that is infinite index in its Zariski closure. We consider the resulting thin orbits of Pythagorean triples O = x0 · Gamma, together with polynomial maps f : O→ Z corresponding to the hypotenuse, area, and product of all three coordinates of these triangles. Let PR be the set of R-almost primes, natural numbers with at most R prime factors. For such an orbit, we say R saturates if the preimage of PR in O under the map f is Zariski dense in O. Denote the minimal R that saturates by R0, the saturation number. We are interested in bounding the saturation number for all Gamma of critical exponent deltaGamma > delta 0, for an explicit delta0. This problem has been of interest since the outset of the affine sieve, and has been studied by Kontorovich [10], Kontorovich-Oh [9], Bourgain-Kontorovich [3], and Hong-Kontorovich [8]. Using an Archimedean sieve and the dispersion method, we improve the best known level of distribution in all three cases. We thereby improve the bounds on the saturation number for areas and product of coordinates, and lower the threshold for the critical exponent delta0 in the instance of hypotenuses.

ISBN: 9780355017762Subjects--Topical Terms:

3173530
Theoretical mathematics.

Almost Primes in Thin Orbits of Pythagorean Triangles.
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300 $a 59 p.
500 $a Source: Dissertation Abstracts International, Volume: 78-11(E), Section: B.
500 $a Adviser: Alex Kontorovich.
502 $a Thesis (Ph.D.)--Yale University, 2017.
506 $a This item is not available from ProQuest Dissertations & Theses.
520 $a Let F = x2 + y2 -- z2, and let x0 epsilon Z 3 be a primitive solution to F(x0) = 0, e.g. so that its coordinates share no nontrivial divisor. Let Gamma ≤ SOF( Z ) be a finitely generated thin subgroup, one that is infinite index in its Zariski closure. We consider the resulting thin orbits of Pythagorean triples O = x0 · Gamma, together with polynomial maps f : O→ Z corresponding to the hypotenuse, area, and product of all three coordinates of these triangles. Let PR be the set of R-almost primes, natural numbers with at most R prime factors. For such an orbit, we say R saturates if the preimage of PR in O under the map f is Zariski dense in O. Denote the minimal R that saturates by R0, the saturation number. We are interested in bounding the saturation number for all Gamma of critical exponent deltaGamma > delta 0, for an explicit delta0. This problem has been of interest since the outset of the affine sieve, and has been studied by Kontorovich [10], Kontorovich-Oh [9], Bourgain-Kontorovich [3], and Hong-Kontorovich [8]. Using an Archimedean sieve and the dispersion method, we improve the best known level of distribution in all three cases. We thereby improve the bounds on the saturation number for areas and product of coordinates, and lower the threshold for the critical exponent delta0 in the instance of hypotenuses.
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650 4 $a Theoretical mathematics. $3 3173530
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