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Alex Kontorovich
Levels of Distribution and the Affine Sieve
Annales de la faculté des sciences de Toulouse Sér. 6, 23 no. 5: Numéro Spécial : Aux croisements de la géométrie hyperbolique et de l’arithmétique (2014), p. 933-966, doi: 10.5802/afst.1432
Article PDF | Reviews MR 3294598

Résumé - Abstract

We discuss the notion of a “Level of Distribution” in two settings. The first deals with primes in progressions, and the role this plays in Yitang Zhang’s theorem on bounded gaps between primes. The second concerns the Affine Sieve and its applications.

Bibliography

[1] Barban (M. B.).— The sieve" method and its application to number theory. Uspehi Mat. Nauk, 21, p. 51-102 (1966).  MR 199171 |  Zbl 0234.10031
[2] Blomer (V.) and Brumley (F.).— On the Ramanujan conjecture over number fields. Ann. of Math. (2), 174(1), p. 581-605 (2011).  MR 2811610
[3] Blomer (V.) and Brumley (F.).— The role of the Ramanujan conjecture in analytic number theory. Bull. Amer. Math. Soc. (N.S.), 50(2), p. 267-320 (2013).  MR 3020828
[4] Bombieri (E.) and Davenport (H.).— Small differences between consecutive prime numbers. Proc. Roy. Soc. Ser. A, p. 1-18 (1966).  MR 199165 |  Zbl 0151.04201
[5] Bombieri (E.), Friedlander (J.), and Iwaniec (H.).— Primes in arithmetic progressions to large moduli. Acta Math., 156, p. 203-251 (1986).  MR 834613 |  Zbl 0588.10042
[6] Bourgain (J.) and Gamburd (A.).— Uniform expansion bounds for Cayley graphs of SL2( Fp). Ann. of Math. (2), 167(2), p. 625-642 (2008).  MR 2415383 |  Zbl 1216.20042
[7] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Sieving and expanders. C. R. Math. Acad. Sci. Paris, 343(3), p. 155-159 (2006).  MR 2246331 |  Zbl 1217.11081
[8] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Affine linear sieve, expanders, and sum-product. Invent. Math., 179(3), p. 559-644 (2010).  MR 2587341 |  Zbl 1239.11103
[9] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Generalization of Selberg’s 3/16th theorem and affine sieve. Acta Math, 207, p. 255-290 (2011).  MR 2892611 |  Zbl 1276.11081
[10] Breuillard (E.), Green (B.), and Tao (T.).— Approximate subgroups of linear groups. Geom. Funct. Anal., 21(4), p. 774-819 (2011).  MR 2827010 |  Zbl 1229.20045
[11] Bourgain (J.) and Kontorovich (A.).— On Zaremba’s conjecture.— Comptes Rendus Mathematique, 349(9), p. 493-495 (2011).  MR 2802911 |  Zbl 1215.11005
[12] Bourgain (J.) and Kontorovich (A.).— On the local-global conjecture for integral Apollonian gaskets (2012). To appear, Invent. Math., arXiv:1205.4416v1, 63 p. 27.  MR 3211042
[13] Bourgain (J.) and Kontorovich (A.).— The affine sieve beyond expansion I: thin hypotenuses (2013). Preprint, arXiv:1307.3535.  MR 3073885
[14] Bourgain (J.) and Kontorovich (A.).— Beyond expansion II: Traces of thin semigroups (2013). Preprint, arXiv:1310.7190.
[15] Bourgain (J.) and Kontorovich (A.).— On Zaremba’s conjecture. Annals Math., 180(1), p. 137-196 (2014).  MR 3194813
[16] Bombieri (E.).— On the large sieve. Mathematika, 12, p. 201-225 (1965).  MR 197425 |  Zbl 0136.33004
[17] Bourgain (J.).— Integral Apollonian circle packings and prime curvatures. J. Anal. Math., 118(1), p. 221-249 (2012).  MR 2993027 |  Zbl 1281.52011
[18] Brun (V.).— Le crible d’Eratosthéne et le theoréme de Goldbach. C. R. Acad. Sci. Paris, 168, p. 544-546 (1919).  JFM 47.0162.01
[19] Burger (M.) and Sarnak (P.).— Ramanujan duals II. Invent. Math, 106, p. 1-11 (1991).  MR 1123369 |  Zbl 0774.11021
[20] Chen (J. R.).— On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16, p. 157-176 (1973).  MR 434997 |  Zbl 0319.10056
[21] Clozel (L.).— Démonstration de la conjecture $\tau $. Invent. Math., 151(2), p. 297-328 (2003).  MR 1953260 |  Zbl 1025.11012
[22] Daveport (H.).— Multiplicative Number Theory, volume 74 of Grad. Texts Math. Springer-Verlag, New York (1980).  MR 606931 |  Zbl 0453.10002
[23] Deuring (M.).— Imaginäre quadratische Zahlkörper mit der Klassenzahl 1. Math. Z., 37(1), p. 405-415 (1933).  MR 1545403 |  Zbl 0007.29602
[24] de la Vallée-Poussin (Ch.J.).— Recherches analytiques sur la théorie des nombers premiers. Ann. Soc. Sci. Bruxelles, 20, p. 183-256 (1896).  JFM 27.0155.03
[25] Elliott (P.D.T.A.) and Halberstam (H.).— A conjecture in prime number theory. Symp. Math. IV (Rome 1968/69), p. 59-72 (1968).  MR 276195 |  Zbl 0238.10030
[26] Erdös (P.).— The difference between consecutive primes. Duke Math J., 6, p. 438-441 (1940).  MR 1759 |  Zbl 0023.29801 |  JFM 66.0162.04
[27] Friedlander (J.) and Granville (A.).— Limitations to the equidistribution of primes. I. Ann. of Math. (2), 129(2), p. 363-382 (1989).  MR 986796 |  Zbl 0671.10041
[28] Fouvry (E.) and Iwaniec (H.).— Primes in arithmetic progressions. Acta Arith., 42, p. 197-218 (1983).  MR 719249 |  Zbl 0517.10045
[29] Friedlander (J.) and Iwaniec (H.).— What is ... the parity phenomenon? Notices Amer. Math. Soc., 56(7), p. 817-818 (2009).  MR 2546824 |  Zbl 1278.11001
[30] Friedlander (J.) and Iwaniec (H.).— Hyperbolic prime number theorem. Acta Math., 202(1), p. 1-19 (2009).  MR 2486486 |  Zbl 1278.11089
[31] Friedlander (J.) and Iwaniec (H.).— Opera de cribro, volume 57 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2010).  MR 2647984 |  Zbl 1226.11099
[32] Friedlander (J.) and Iwaniec (H.).— Close encounters among the primes (2014). arXiv:1312.2926.  MR 3286081
[33] Frolenkov (D.) and Kan (I. D.).— A reinforcement of the Bourgain-Kontorovich’s theorem by elementary methods II (2013). Preprint, arXiv:1303.3968.
[34] Fuchs (E.), Meiri (C.), and Sarnak (P.).— Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, 2012. To appear, JEMS.  MR 3262453
[35] Fouvry (E.).— Autour du théorème de Bombieri-Vinogradov. Acta Math, 152, p. 219-244 (1984).  MR 741055 |  Zbl 0552.10024
[36] Fuchs (E.).— The ubiquity of thin groups (2012). To appear, MSRI Proceedings.  MR 3220885
[37] Gamburd (A.).— On the spectral gap for infinite index “congruence" subgroups of SL2(Z). Israel J. Math., 127, p. 157-200 (2002).  MR 1900698 |  Zbl 1028.11031
[38] Goldston (D. A.), Graham (S. W.), Pintz (J.), and Yildirim (C. Y.).— Small gaps between products of two primes. Proc. London Math. Soc., 98(3), p. 741-774 (2009).  MR 2500871 |  Zbl 1213.11171
[39] Goldfeld (D.).— The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3(4), p. 624-663 (1976). Numdam |  MR 450233 |  Zbl 0345.12007
[40] Goldfeld (D.).— Gauss’s class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc. (N.S.), 13(1), p. 23-37 (1985).  MR 788386 |  Zbl 0572.12004
[41] Goldston (D. A.).— On Bombieri and Davenport’s theorem concerning small gaps between primes. Mathematika, 39(1), p. 10-17 (1992).  MR 1176465 |  Zbl 0758.11037
[42] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— The path to recent progress on small gaps between primes. In Analytic number theory, volume 7 of Clay Math. Proc., pages 129-139. Amer. Math. Soc., Providence, RI (2007).  MR 2362197 |  Zbl 1213.11168
[43] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— Primes in tuples I. Ann. of Math. (2), 170(2), p. 819-862 (2009).  MR 2552109 |  Zbl 1207.11096
[44] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— Primes in tuples II. Acta Math., 204, p. 1-47 (2010).  MR 2600432 |  Zbl 1207.11097
[45] Granville (A.).— Harald Cramér and the distribution of prime numbers. Scand. Actuar. J., (1), p. 12-28 (1995). Harald Cramér Symposium (Stockholm, 1993).  MR 1349149 |  Zbl 0833.01018
[46] Green (B.).— Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott, and Sarnak. Current Events Bulletin, AMS (2010).
[47] Green (B.) and Tao (T.).— Linear equations in primes. Ann. of Math. (2), 171(3), p. 1753-1850 (2010).  MR 2680398 |  Zbl 1242.11071
[48] Gross (B. H.) and Zagier (D. B).— Heegner points and derivatives of L-series. Invent. Math., 84(2), p. 225-320 (1986).  MR 833192 |  Zbl 0608.14019
[49] Hadamard (J.).— Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France, 24, p. 199-220 (1896).  MR 1504264 |  JFM 27.0154.01
[50] Heath-Brown (D. R.).— Prime twins and Siegel zeros. Proc. London Math. Soc. (3), 47(2), p. 193-224 (1983).  MR 703977 |  Zbl 0517.10044
[51] Heilbronn (H.).— On the class number in imaginary quadratic elds. Quarterly J. of Math., 5, p. 150-160 (1934).  Zbl 0009.29602 |  JFM 60.0155.01
[52] Helfgott (H. A.).— Growth and generation in $SL_2(\mathbb{Z}/p \mathbb{Z})$. Ann. of Math. (2), 167(2), p. 601-623 (2008).  MR 2415382 |  Zbl 1213.20045
[53] Hong (J.) and Kontorovich (A.).— Almost prime coordinates for anisotropic and thin Pythagorean orbits (2014). To appear, Israel J. Math. arXiv:1401.4701.
[54] Hoory (S.), Linial (N.), and Wigderson (A.).— Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4), p. 439-561 (electronic), 2006.  MR 2247919 |  Zbl 1147.68608
[55] Huang (S.).— An improvement on Zaremba’s conjecture (2013). Preprint, arXiv:1310.3772.
[56] Huxley (M. N.).— Small differences between consecutive primes. II. Mathematika, 24, p. 142-152 (1977).  MR 466042 |  Zbl 0367.10038
[57] Iwaniec (H.) and Kowalski (E.).— Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004.  MR 2061214 |  Zbl 1059.11001
[58] Kim (H. H.).— Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. J. Amer. Math. Soc., 16(1), p. 139-183 (electronic) (2003). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Sarnak (P.).  MR 1937203 |  Zbl 1018.11024
[59] Kontorovich (A.) and Oh (H.).— Almost prime Pythagorean triples in thin orbits. J. reine angew. Math., 667, p. 89-131 (2012). arXiv:1001.0370.  MR 2929673 |  Zbl 1273.11055
[60] Kontorovich (A. V.).— The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves. Columbia University Thesis (2007).  MR 2710911
[61] Kontorovich (A.).— The hyperbolic lattice point count in infinite volume with applications to sieves. Duke J. Math., 149(1), p. 1-36 (2009). arXiv:0712.1391.  MR 2541126 |  Zbl 1223.11113
[62] Kontorovich (A.).— Expository note: an arithmetic surface (2011). Unpublished note, http://math.yale.edu/   avk23/files/UniformLattice. pdf.
[63] Kontorovich (A.).— From Apollonius to Zaremba: local-global phenomena in thin orbits. Bull. Amer. Math. Soc. (N.S.), 50(2), p. 187-228 (2013).  MR 3020826
[64] Kowalski (E.).— Sieve in expansion. Séminaire Bourbaki, 63(1028), p. 1-35 (2011).
[65] Landau (E.).— Über die Klassenzahl imaginär-quadratischer Zahlkörper. Nachr. Ges. Wiss. Gottingen, p. 285-295 (1918).  JFM 46.0258.04
[66] Landau (E.).— Bemerkungen zum Heilbronnschen Satz. Acta Arith, p. 1-18 (1935).  JFM 61.0170.01
[67] Linnik (U. V.).— The large sieve. C. R. (Doklady) Acad. Sci. URSS (N.S.), 30, p. 292-294 (1941).  MR 4266 |  JFM 67.0128.01
[68] Lax (P.D.) and Phillips (R.S.).— The asymptotic distribution of lattice points in Euclidean and non-Euclidean space. Journal of Functional Analysis, 46, p. 280-350 (1982).  MR 661875 |  Zbl 0497.30036
[69] Luo (W.), Rudnick (Z.), and Sarnak (P.).— On Selberg’s eigenvalue conjecture. Geom. Funct. Anal., 5(2), p. 387-401 (1995).  MR 1334872 |  Zbl 0844.11038
[70] Liu (J.) and Sarnak (P.).— Integral points on quadrics in three variables whose coordinates have few prime factors. Israel J. Math, 178, p. 393-426 (2010).  MR 2733075 |  Zbl 1230.11045
[71] Lubotzky (A.).— Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc., 49, p. 113-162 (2012).  MR 2869010 |  Zbl 1232.05194
[72] Maier (H.).— Primes in short intervals. Michigan Math. J., 32(2), p. 221-225 (1985).  MR 783576 |  Zbl 0569.10023
[73] Maier (H.).— Small differences between prime numbers. Michigan Math J., 35, p. 323-344 (1988).  MR 978303 |  Zbl 0671.10037
[74] Maynard (J.).— Small gaps between primes (2013). Preprint, arXiv:1311.4600.
[75] McMullen (C. T.).— Uniformly Diophantine numbers in a fixed real quadratic field. Compos. Math., 145(4), p. 827-844 (2009).  MR 2521246 |  Zbl 1176.11032
[76] McMullen (C. T.).— Dynamics of units and packing constants of ideals, 2012. Online lecture notes, http://www.math.harvard.edu/   ctm/ expositions/home/text/papers/cf/slides/slides.pdf.
[77] Montgomery (H. L.).— Topics in Multiplicative Number Theory, volume 227 of Lecture Notes in Math. Springer, New York (1971).  MR 337847 |  Zbl 0216.03501
[78] Mordell (L. J.).— On the riemann hypothesis and imaginary quadratic fields with a given class number. J. London Math. Soc., 9, p. 289-298 (1934).  MR 1574881 |  Zbl 0010.24902
[79] Motohashi (Y) and Pintz (J.).— A smoothed GPY sieve. Bull. Lond. Math. Soc., 40(2), p. 298-310 (2008).  MR 2414788 |  Zbl 1278.11090
[80] Novikov (P. S.).— Ob algoritmiěskoǐ nerazrešimosti problemy toždestva slov v teorii grupp. Trudy Mat. Inst. im. Steklov. no. 44. Izdat. Akad. Nauk SSSR, Moscow (1955).  MR 75197 |  Zbl 0068.01301
[81] Nevo (A.) and Sarnak (P.).— Prime and almost prime integral points on principal homogeneous spaces (2009).  MR 2746350 |  Zbl 1233.11102
[82] Pyber (L.) and Szabo (E.).— Growth in finite simple groups of lie type of bounded rank, 2010. Preprint arXiv:1005.1858.
[83] Rényi (A.).— On the representation of an even number as the sum of a single prime and single almost-prime number. Izvestiya Akad. Nauk SSSR. Ser. Mat., 12, p. 57-78 (1948).  MR 23863 |  Zbl 0038.18601
[84] Rankin (R. A.).— The difference between consecutive prime numbers. II. Proc. Cambridge Philos. Soc., 36, p. 255-266 (1940).  MR 1760 |  JFM 66.0163.01
[85] Riemann (B.).— Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie (1859).
[86] Roth (K.F.).— On the large sieves of Linnik and Rényi. Mathematika, 12, p. 1-9 (1965).  MR 197424 |  Zbl 0137.25904
[87] Sarnak (P.).— Selberg’s eigenvalue conjecture. Notices Amer. Math. Soc., 42(11), p. 1272-1277 (1995).  MR 1355461 |  Zbl 1042.11517
[88] Sarnak (P.).— What is... an expander? Notices Amer. Math. Soc., 51(7), p. 762-763 (2004).  MR 2072849 |  Zbl 1161.05341
[89] Sarnak (P.).— Notes on the generalized Ramanujan conjectures. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 659-685. Amer. Math. Soc., Providence, RI (2005).  MR 2192019 |  Zbl 1146.11031
[90] Sarnak (P.).— Letter to J. Lagarias (2007). http://web.math.princeton. edu/sarnak/AppolonianPackings.pdf.  MR 2362203
[91] Sarnak (P.).— Equidistribution and primes. Astérisque, (322), p. 225-240 (2008). Géométrie différentielle, physique mathématique, mathématiques et société. II.  MR 2521658 |  Zbl 1223.11112
[92] Sarnak (P.).— Affine sieve (2010). Slides from lectures, http://www.math. princeton.edu/sarnak/Affinesievesummer2010.pdf.
[93] Sarnak (P.).— Notes on thin matrix groups. In Thin Groups and Superstrong Approximation, volume 61 of Mathematical Sciences Research Institute Publications, p. 343-362. Cambridge University Press (2014).  MR 3220897
[94] Selberg (A.).— On the estimation of Fourier coefficients of modular forms. Proc. of Symposia in Pure Math., VII, p. 1-15 (1965).  MR 182610 |  Zbl 0142.33903
[95] Serre (J.-P.).— Topics in Galois theory, volume 1 of Res. Notes in Math. A.K. Peters (2008).  MR 2363329 |  Zbl 1128.12001
[96] Salehi Golsefidy (A.).— Affine sieve and expanders (2012). To appear, Proceedings of MSRI.
[97] Salehi Golsefidy (A.) and Sarnak (P.).— Affine sieve (2011). To appear, JAMS.
[98] Salehi Golsefidy (A.) and Varjú (P. P.).— Expansion in perfect groups. Geom. Funct. Anal., 22(6), p. 1832-1891 (2012).  MR 3000503 |  Zbl 1284.20044
[99] Siegel (C. L.).— Über die Classenzahl quadratischer Zahlkörper. Acta Arith, 1, p. 83-86 (1935).  JFM 61.0170.02
[100] Soundararajan (K.).— Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim. Bull. Amer. Math. Soc. (N.S.), 44(1), p. 1-18 (2007).  MR 2265008 |  Zbl 1193.11086
[101] Sarnak (P.) and Xue (X.).— Bounds for multiplicities of automorphic representations. Duke J. Math., 64(1), p. 207-227 (1991).  MR 1131400 |  Zbl 0741.22010
[102] Vinogradov (A. I.).— The density hypothesis for Dirichet L-series. Izv. Akad. Nauk SSSR Ser. Mat., 29, p. 903-934, 1965.  MR 197414 |  Zbl 0128.04205
[103] Walfisz (A.).— Zur additiven Zahlentheorie. II. Math. Z., 40(1), p. 592-607 (1936).  MR 1545584 |  JFM 61.1070.02
[104] Zhang (Y.).— Bounded gaps between primes (2013). To appear, Annals Math. 2  MR 3171761 |  Zbl 1290.11128
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