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Almost Primes in Thin Orbits of Pythagorean Triangles.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Almost Primes in Thin Orbits of Pythagorean Triangles./
Author:	Ehrman, Max.
Description:	1 online resource (59 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 78-11(E), Section: B.
Contained By:	Dissertation Abstracts International78-11B(E).
Subject:	Theoretical mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355017762

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

Almost Primes in Thin Orbits of Pythagorean Triangles. - 1 online resource (59 pages)

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

Thesis (Ph.D.)

Includes bibliographical references

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.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355017762Subjects--Topical Terms:

1180455
Theoretical mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

Almost Primes in Thin Orbits of Pythagorean Triangles.
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500 $a Source: Dissertation Abstracts International, Volume: 78-11(E), Section: B.
500 $a Adviser: Alex Kontorovich.
502 $a Thesis (Ph.D.) $c Yale University $d 2017.
504 $a Includes bibliographical references
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.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Theoretical mathematics. $3 1180455
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
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710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a Yale University. $3 1178968
773 0 $t Dissertation Abstracts International $g 78-11B(E).
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