Affine linear sieve, expanders, and sum-product

By Jean Bourgain, Alex Gamburd, Peter Sarnak. 2010

203 citations recorded by the source corpus.

AbstractLet $\mathcal{O}$ be an orbit in ℤn of a finitely generated subgroup Λ of GLn(ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$ . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/031282963fd7a47c18b6266d06667ad99e25bf71

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In Algebra and Number Theory

1. H. Oh
2. Alex Kontorovich
3. A. Mohammadi
4. Peter Sarnak
5. A. Nevo
6. P. P. Varj'u
7. László Pyber
8. E. Szab'o

In Discrete Mathematics and Combinatorics

1. Alex Kontorovich
2. H. Oh
3. A. Mohammadi
4. P. P. Varj'u
5. A. Nevo
6. Peter Sarnak
7. E. Szab'o
8. László Pyber

Citations in the corpus

• Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, methodology): “For the principal congruence subgroup The following is obtained by Bourgain, Gamburd and Sarnak in [4] and [5].”
• Random walks in compact groups (2012, background): “The statement of the following theorem is very technical, so we first explain it informally: We suppose that G is a quasirandom profinite group and partition its finite index normal subgroups into two sets: Ω1∪̇Ω2.”
• Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “In both theorems above, if all BT are K-invariant subsets, our hypothesis on the uniform spectral gap for the family {0d} can be disposed of again, as the uniform spherical spectral gap property proved in [58] and [6] is sufficient in this case.”
• Growth in finite simple groups of Lie type of bounded rank (2010, methodology): “This result was used in [7] as a building block in a combinatorial sieve method for primes and almost primes on orbits of various subgroups of SL(2,Z) as they act on Z (for m ≥ 2).”