Non-commutative methods in additive combinatorics and number theory
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Austin, 2015, Quantitative equidistribution for certain quadruples in quasi-random groups, Combin. Probab. Comput., 24, 376, 10.1017/S0963548314000492
Austin, 2016, Ajtai–Szemerédi theorems over quasirandom groups, Recent trends in combinatorics, 159, 453, 10.1007/978-3-319-24298-9_19
Babai, 1992, On the diameter of permutation groups, European J. Combin., 13, 231, 10.1016/S0195-6698(05)80029-0
Borthwick, 2007, Progr. Math., 256
Bourgain, 2012, A modular Szemerédi–Trotter theorem for hyperbolas, C. R. Math. Acad. Sci. Paris, 350, 793, 10.1016/j.crma.2012.09.011
Bourgain, 2014, Some Diophantine applications of the theory of group expansion, Thin groups and superstrong approximation, 61, 1
Bourgain, 2008, Uniform expansion bounds for Cayley graphs of $\operatorname{SL}_2(\mathbb{F}_p)$, Ann. of Math. (2), 167, 625, 10.4007/annals.2008.167.625
Bourgain, 2010, Affine linear sieve, expanders, and sum-product, Invent. Math., 179, 559, 10.1007/s00222-009-0225-3
Bourgain, 2006, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2), 73, 380, 10.1112/S0024610706022721
Bourgain, 2011, On Zaremba’s conjecture, C. R. Math. Acad. Sci. Paris, 349, 493, 10.1016/j.crma.2011.03.023
Bourgain, 2014, On the local–global conjecture for integral Apollonian gaskets, Invent. Math., 196, 589, 10.1007/s00222-013-0475-y
Breuillard, 2014, A brief introduction to approximate groups, Thin groups and superstrong approximation, 61, 23
Breuillard, 2011, Approximate groups. II. The solvable linear case, Q. J. Math., 62, 513, 10.1093/qmath/haq011
Breuillard, 2011, Approximate subgroups of linear groups, Geom. Funct. Anal., 21, 774, 10.1007/s00039-011-0122-y
Breuillard, 2012, The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci., 116, 115, 10.1007/s10240-012-0043-9
Brun, 1920, Skr. Nor. Vidensk. Akad. Kristiania I, 3
Button, 2015, An explicit upper bound for the Helfgott delta in $\operatorname{SL}(2,p)$, J. Algebra, 421, 493, 10.1016/j.jalgebra.2014.09.001
DeVos, 2013
Elekes, 1997, On linear combinatorics. I. Concurrency – an algebraic approach, Combinatorica, 17, 447, 10.1007/BF01194999
Elekes, 1998, On linear combinatorics. II. Structure theorems via additive number theory, Combinatorica, 18, 13, 10.1007/PL00009806
Elekes, 2002, SUMS versus PRODUCTS in number theory, algebra and Erdős geometry, Paul Erdős and his mathematics, 11, 241
Eskin, 2005, On uniform exponential growth for linear groups, Invent. Math., 160, 1, 10.1007/s00222-004-0378-z
Fouvry, 2014, Algebraic trace functions over the primes, Duke Math. J., 163, 1683, 10.1215/00127094-2690587
Freiman, 1973, Groups and inverse problems in additive number theory, Number-theoretic studies on Markov spectrum and the structure theory of set addition, 175
Freiman, 1966, Transl. Math. Monogr., 37
Freiman, 2012, On finite subsets of nonabelian groups with small doubling, Proc. Amer. Math. Soc., 140, 2997, 10.1090/S0002-9939-2012-11156-6
Frobenius, 1896, Über Gruppencharaktere, Sitzungber. Preuss. Akad. Wiss. Berlin, 985
Frolenkov, 2014, A strengthening of a theorem of Bourgain– Kontorovich. II, Mosc. J. Comb. Number Theory, 4, 78
Gamburd, 2002, On the spectral gap for infinite index ‘congruence’ subgroups of $\operatorname{SL}_2(\mathbb Z)$, Israel J. Math., 127, 157, 10.1007/BF02784530
Gill, 2013, On the product decomposition conjecture for finite simple groups, Groups Geom. Dyn., 7, 867, 10.4171/GGD/208
Golsefidy, 2012, Expansion in perfect groups, Geom. Funct. Anal., 22, 1832, 10.1007/s00039-012-0190-7
Gowers, 1998, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal., 8, 529, 10.1007/s000390050065
Green, 2007, Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2), 75, 163, 10.1112/jlms/jdl021
Gromov, 1981, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53, 53, 10.1007/BF02698687
Guth, 2015, On the Erdős distinct distances problem in the plane, Ann. of Math. (2), 181, 155, 10.4007/annals.2015.181.1.2
Hamidoune, 2008, Some additive applications of the isoperimetric approach, Ann. Inst. Fourier (Grenoble), 58, 2007, 10.5802/aif.2404
Hardy, 1923, Some problems of ‘Partitio Numerorum’. III. On the expression of a number as a sum of primes, Acta Math., 44, 1, 10.1007/BF02403921
Helfgott, 2008, Growth and generation in $\operatorname{SL}_2(\mathbb{Z}/p\mathbb{Z})$, Ann. of Math. (2), 167, 601, 10.4007/annals.2008.167.601
Helfgott, 2011, Growth in $\operatorname{SL}_3(\mathbb{Z}/p\mathbb{Z})$, J. Eur. Math. Soc. (JEMS), 13, 761, 10.4171/JEMS/267
Helfgott, 2015, Growth in groups: ideas and perspectives, Bull. Amer. Math. Soc. (N. S.), 52, 357, 10.1090/S0273-0979-2015-01475-8
Hensley, 1989, The distribution of badly approximable numbers and continuants with bounded digits, Théorie des nombres, 371, 10.1515/9783110852790.371
Hensley, 1990, The distribution of badly approximable rationals and continuants with bounded digits. II, J. Number Theory, 34, 293, 10.1016/0022-314X(90)90139-I
Hensley, 1992, Continued fraction Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory, 40, 336, 10.1016/0022-314X(92)90006-B
Hensley, 1994, The distribution mod $n$ of fractions with bounded partial quotients, Pacific J. Math., 166, 43, 10.2140/pjm.1994.166.43
Hensley, 1996, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Number Theory, 58, 9, 10.1006/jnth.1996.0058
Hlawka, 1961, Funktionen von beschränkter Variation in der Theorie der Gleichverteilung, Ann. Mat. Pura Appl. (4), 54, 325, 10.1007/BF02415361
Hrushovski, 2012, Stable group theory and approximate subgroups, J. Amer. Math. Soc., 25, 189, 10.1090/S0894-0347-2011-00708-X
Hrushovski, 2008, Counting and dimensions, Model theory with applications to algebra and analysis, 350, 161, 10.1017/CBO9780511735219.005
Huxley, 1986, Exceptional eigenvalues and congruence subgroups, The Selberg trace formula and related topics, 53, 341, 10.1090/conm/053/853564
Kan, 2016, A strengthening of a theorem of Bourgain and Kontorovich. IV, Izv. Ross. Akad. Nauk Ser. Mat., 80, 103, 10.4213/im8360
Kempermann, 1956, On complexes in a semigroup, Nederl. Akad. Wetensch. Proc. Ser. A, 59, 247, 10.1016/S1385-7258(56)50032-7
Kerr, 2020, Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progression, Res. Number Theory, 6, 10.1007/s40993-020-0191-9
Khinchin, 1961
Koksma, 1942, A general theorem from the theory of uniform distribution modulo $1$, Mathematica, Zutphen. B, 11, 7
Kontorovich, 2007
Kontorovich, 2013, From Apollonius to Zaremba: local–global phenomena in thin orbits, Bull. Amer. Math. Soc. (N. S.), 50, 187, 10.1090/S0273-0979-2013-01402-2
Kontorovich, 2014, The orbital circle method, Thin groups and superstrong approximation, 61, 93
Kontorovich, 2014, Levels of distribution and the affine sieve, Ann. Fac. Sci. Toulouse Math. (6), 23, 933, 10.5802/afst.1432
Korobov, 1963
Kowalski, 2013, Explicit growth and expansion for $\operatorname{SL}_2$, Int. Math. Res. Not. IMRN, 2013, 5645, 10.1093/imrn/rns214
Kowalski, 2019, Cours Spéc., 26
Kowalski, 2017, Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2), 186, 413, 10.4007/annals.2017.186.2.2
Kowalski, 2020
Kuipers, 1974, Pure Appl. Math.
Landazuri, 1974, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra, 32, 418, 10.1016/0021-8693(74)90150-1
Lang, 1954, Number of points of varieties in finite fields, Amer. J. Math., 76, 819, 10.2307/2372655
Larsen, 2011, Finite subgroups of algebraic groups, J. Amer. Math. Soc., 24, 1105, 10.1090/S0894-0347-2011-00695-4
Linnik, 1967, Ergeb. Math. Grenzgeb., 45
Lubotzky, 1995, Cayley graphs: eigenvalues, expanders and random walks, Surveys in combinatorics, 218, 155, 10.1017/CBO9780511662096.008
Macourt, 2019, Double sums of Kloosterman sums in finite fields, Finite Fields Appl., 60, 10.1016/j.ffa.2019.101575
Magee, 2015
Magee, 2019, Uniform congruence counting for Schottky semigroups in $\operatorname{SL}_2 (\mathbb{Z})$, J. Reine Angew. Math., 2019, 89, 10.1515/crelle-2016-0072
Margulis, 1982, Explicit constructions of graphs without short cycles and low density codes, Combinatorica, 2, 71, 10.1007/BF02579283
Moshchevitin, 2007, Sets of the form $\mathscr A+\mathscr B$ and finite continued fractions, Mat. Sb., 198, 95, 10.4213/sm3772
Moshchevitin, 2012
Moshchevitin, 2020, Popular products and continued fractions, Israel J. Math., 238, 807, 10.1007/s11856-020-2039-3
Moshchevitin, 2020, On a modular form of Zaremba’s conjecture, Pacific J. Math., 309, 195, 10.2140/pjm.2020.309.195
Murphy, 2019
Murphy, 2021, Upper and lower bounds for rich lines in grids, Amer. J. Math., 143, 577, 10.1353/ajm.2021.0013
Murphy, 2015, Variations on the sum-product problem, SIAM J. Discrete Math., 29, 514, 10.1137/140952004
Murphy, 2019
Murphy, 2020, Growth in some finite three-dimensional matrix groups, SIAM J. Discrete Math., 34, 1984, 10.1137/20M1338265
Naimark, 2010, Grundlehren Math. Wiss., 246
Niederreiter, 1986, Dyadic fractions with small partial quotients, Monatsh. Math., 101, 309, 10.1007/BF01559394
Nikolov, 2011, Product decompositions of quasirandom groups and a Jordan type theorem, J. Eur. Math. Soc. (JEMS), 13, 1063, 10.4171/JEMS/275
Noether, 1922, Ein algebraisches Kriterium für absolute Irreduzibilität, Math. Ann., 85, 26, 10.1007/BF01449599
Ol’mezov, 2021, An elementary analog of the operator method in additive combinatorics, Mat. Zametki, 109, 117, 10.4213/mzm12764
Olson, 1984, On the sum of two sets in a group, J. Number Theory, 18, 110, 10.1016/0022-314X(84)90047-7
Pach, 2004, Geometric incidences, Towards a theory of geometric graphs, 342, 185, 10.1090/conm/342/06151
Petridis, 2012, New proofs of Plünnecke-type estimates for product sets in groups, Combinatorica, 32, 721, 10.1007/s00493-012-2818-5
Petridis, 2022, An energy bound in the affine group, Int. Math. Res. Not. IMRN, 2022, 1154, 10.1093/imrn/rnaa130
Pollicott, 2020
Pyber, 2014, Growth in linear groups, Thin groups and superstrong approximation, 61, 253
Pyber, 2016, Growth in finite simple groups of Lie type, J. Amer. Math. Soc., 29, 95, 10.1090/S0894-0347-2014-00821-3
Rapinchuk, 2014, Strong approximation for algebraic groups, Thin groups and superstrong approximation, 61, 269
Razborov, 2014, A product theorem in free groups, Ann. of Math. (2), 179, 405, 10.4007/annals.2014.179.2.1
Roche-Newton, 2019, On the size of the set $AA+A$, J. Lond. Math. Soc. (2), 99, 477, 10.1112/jlms.12177
Roche-Newton, 2021, New expander bounds from affine group energy, Discrete Comput. Geom., 66, 552, 10.1007/s00454-020-00194-z
Rudnev, 2018, On the number of incidences between points and planes in three dimensions, Combinatorica, 38, 219, 10.1007/s00493-016-3329-6
Rudnev, 2019 (v1 – 2018)
Rukavishnikova, 2006, A probability estimate for the sum of incomplete partial quotients with fixed denominator, Chebyshevskii Sb., 7, 113
Ruzsa, Sums of finite sets, Number theory: New York seminar 1991–1995, 281, 10.1007/978-1-4612-2418-1_21
Sanders, 2011, A quantitative version of the non-Abelian idempotent theorem, Geom. Funct. Anal., 21, 141, 10.1007/s00039-010-0107-2
Sanders, 2012, Approximate groups and doubling metrics, Math. Proc. Cambridge Philos. Soc., 152, 385, 10.1017/S0305004111000740
Sanders, 2012
Sanders, 2013, The structure theory of sets addition revisited, Bull. Amer. Math. Soc. (N. S.), 50, 93, 10.1090/S0273-0979-2012-01392-7
Sarnak, 1991, Bounds for multiplicities of automorphic representations, Duke Math. J., 64, 207, 10.1215/S0012-7094-91-06410-0
Schinzel, 1958, Sur certaines hypothèses concernant les nombres premiers, Acta Arith., 4, 185, 10.4064/aa-4-3-185-208
Schoen, 2013, Higher moments of convolutions, J. Number Theory, 133, 1693, 10.1016/j.jnt.2012.10.010
Serre, 1967, Collection Méthodes
Shkredov, 2013, Some new results on higher energies, Trans. Moscow Math. Soc., 74, 35, 10.1090/S0077-1554-2014-00212-0
Shkredov, 2015, Structure theorems in additive combinatorics, Uspekhi Mat. Nauk, 70, 123, 10.4213/rm9647
Shkredov, 2017, Some remarks on the Balog–Wooley decomposition theorem and quantities $D^{+}$, $D^\times$, Proc. Steklov Inst. Math., 24, 85, 10.4213/spm71
Shkredov, 2018, On asymptotic formulae in some sum-product questions, Trans. Moscow Math. Soc., 79, 271, 10.1090/mosc/283
Shkredov, 2020, Some remarks on products of sets in the Heisenberg group and in the affine group, Forum Math., 32, 189, 10.1515/forum-2019-0181
Shkredov, 2020
Shkredov, 2021, On the spectral gap and the diameter of Cayley graphs, Tr. Mat. Inst. Steklova, 314, 318, 10.4213/tm4199
Shkredov, 2021, Modular hyperbolas and bilinear forms of Kloosterman sums, J. Number Theory, 220, 182, 10.1016/j.jnt.2020.06.014
Shparlinski, 2008, Distribution of modular inverses and multiples of small integers and the Sato–Tate conjecture on average, Michigan Math. J., 56, 99, 10.1307/mmj/1213972400
Shparlinski, 2019, On sums of Kloosterman and Gauss sums, Trans. Amer. Math. Soc., 371, 8679, 10.1090/tran/7506
Shparlinski, 2016, Cancellations amongst Kloosterman sums, Acta Arith., 176, 201, 10.4064/aa8365-6-2016
Steinberg, 1968
Stevens, 2017, An improved point-line incidence bound over arbitrary fields, Bull. Lond. Math. Soc., 49, 842, 10.1112/blms.12077
Tao, 2008, Product set estimates for non-commutative groups, Combinatorica, 28, 547, 10.1007/s00493-008-2271-7
Tointon, 2014, Freiman’s theorem in an arbitrary nilpotent group, Proc. Lond. Math. Soc. (3), 109, 318, 10.1112/plms/pdu005
Vinogradov, 2004
Zaremba, 1966, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. (4), 73, 293, 10.1007/BF02415091