2013/07/12 by Jean Bourgain, Bourgain, Jean, Alex Kontorovich +1
Mathematics · #11E25 #11F72 #11N36 #20H10 #22E40 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11E25 #msc:11F72 #msc:11N36 #msc:20H10 #msc:22E40
paper · pdf · doi:10.48550/arxiv.1307.3535
33 pages, 1 figure
arxiv created 2013/07/12 · openalex publication_date 2013/07/12 · arxiv updated 2013/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha < 1/4, and R = 5 almost primes. The exponent 1/4 is the natural analogue of the "Bombieri-Vinogradov" range of distribution for this problem, see Remark 1.19. In this paper, we unconditionally prove the exponent alpha < 7/24 (in the "Elliott-Halberstam" range), thereby producing R = 4 almost primes. The main tools involve developing bilinear forms and the dispersion method in the range of incomplete sums for this Affine Sieve problem.