Research Interests
I am a mathematician specializing in Additive Combinatorics,
Number Theory, Classical Analysis, and Combinatorial Ergodic Theory.
Research topics:
exponential sums and Fourier analysis, the sum-product phenomenon,
inverse results in additive combinatorics, uniformly distributed sequences, continued fractions, ordinary and multiple recurrence, multi-dimensional generalizations of Szemeredi's theorem
cv |
surveys |
papers |
teaching
Education & Degrees
- Doctor of Physical and Math. Sciences (Habilitation), Saint-Petersburg State University, 2009.
- Candidate of Sciences (PhD), Moscow State University, 2005.
Advisor: Professor Nikolay G. Moshchevitin
Past Fellowships & Visiting Positions
- Postdoctoral Member, IAS, Princeton (Arithmetic Combinatorics program, 2007)
- Postdoctoral Member, MSRI, Berkeley (Dynamical Systems and Additive Combinatorics program, 2008)
- Consultant, Microsoft Research (five terms)
- Visiting Professor / Researcher at MPIM (Bonn), University of Sydney/Mathematical Research Institute, LIMS,
Mathematisches Institut (Gottingen).
Noncommutative methods in Additive Combinatorics and Number Theory.
Uspekhi Mat. Nauk, 76:6 (462) (2021) 119–180
Structural theorems in additive combinatorics.
Russian Mathematical Surveys, 70:1 (2015) 123–178
Fourier analysis in combinatorial number theory.
Russian Mathematical Surveys, 65:3 (2010) 513–567
Szemerédi’s theorem and problems concerning arithmetic progressions.
Russian Mathematical Surveys, 61:6 (2006) 1101–1166
Selected recent papers & preprints
(some old papes can be found on) arXiv author page.
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On the Balog-Szemeredi-Gowers theorem for real numbers.
Acta Math. Hungarica (submitted).
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On new configurations in sumsets.
(with C. Elsholtz and L. Wurzinger)
Mathematika (submitted).
- Structure theory of set addition with two operations. (with A. Semchankau)
Combinatorica (submitted).
- On common energies and sumsets II.
Discrete Math. (submitted).
- On common energies and sumsets.
JCTA, 213 (2025) 106026; DOI:10.1016/j.jcta.2025.106026
- On the distribution of quadratic residues.
FFA, 103 (2025) 102577, 1–19
- Energy bounds for modular roots and their applications. (with B. Kerr, I.E. Shparlinski and A. Zaharescu)
Journal of the Institute of Mathematics of Jussieu, (2024) 1–42; DOI:10.1017/S1474748023000397
- Additive dimension and the growth of sets.
Discrete Mathematics, 347:9 (2024) 114077; DOI:10.1016/j.disc.2024.114077
- On the distribution of modular square roots of primes. (with I.E. Shparlinski and A. Zaharescu)
Mathematische Zeitschrift, 306:43 (2024) 1–17; DOI:10.1007/s00209-024-03436-5
- Bilinear forms with modular square roots and averages of twisted second moments of half integral weight Dirichlet series. (with I.E. Shparlinski and A. Zaharescu)
International Mathematics Research Notices, Volume 2022, Issue 22 (2022) 17431–17474; DOI:10.1093/imrn/rnab220
- The Uniformity Conjecture in Additive Combinatorics. (with J. Solymosi)
SIAM Journal on Discrete Mathematics, 35:1 (2021) 307–321
- Tilted Corners in Integer Grids. (with J. Solymosi)
INTEGERS Journal, Ron Graham Memorial Volume 21A (2021), A20
- On the spectral gap and the diameter of Cayley graphs.
Proceedings of the Steklov Institute of Mathematics, 314 (2021) 318–337
- L-functions and sum-free sets. (with T. Schoen)
Acta Mathematica Hungarica, 161:2 (2020) 427–442
- A remark on sets with small Wiener norm.
In: Trigonometric Sums and Their Applications, Springer, Cham (2020) 261–272; DOI:10.1007/978-3-030-37904-9_12
- An upper bound for weak Bk-sets. (with T. Schoen)
SIAM Journal on Discrete Mathematics, 33:2 (2019) 837–844; DOI:10.1137/18M1189439
- A short note on the multiplicative energy of the spectrum of a set.
Mathematical Notes, 105:3 (2019) 444–454
- On subgraphs of random Cayley sum graphs. (with S.V. Konyagin)
European Journal of Combinatorics, 70 (2018) 61–74
- Packing sets over finite abelian groups. (with O. Roche-Newton and A. Winterhof)
INTEGERS, 18 (2018) 1–9
- A remark on A+B and A−A for compact sets in Rⁿ. (with T. Schoen)
INTEGERS, 16 (2016) A75
- Additive dimension and a theorem of Sanders. (with T. Schoen)
Journal of the Australian Mathematical Society, 100:1 (2016) 124–144; DOI:10.1017/S1446788715000324
- On Wiener norm of subsets of Z_p of medium size. (with S.V. Konyagin)
Fundamenta et Applicandae Mathematicae, 19:5 (2014) 75–87
- Sumsets in quadratic residues.
Acta Arithmetica, 164:3 (2014) 221–244
- Roth’s theorem in many variables. (with T. Schoen)
Israel Journal of Mathematics, 199:1 (2014) 287–308
- Sets with large additive energy and symmetric sets. (with S. Yekhanin)
Journal of Combinatorial Theory, Series A, 118 (2011) 1086–1093
- Fourier analysis in combinatorial number theory.
Russian Mathematical Surveys, 65:3 (2010) 513–567
- On monochromatic solutions of some nonlinear equations in Z/pZ.
Mathematical Notes, 88:4 (2010) 626–635
- On a result of J. Bourgain. (with S.V. Konyagin)
Ukrainian Mathematical Journal, 62:3 (2010) 332–368
- On the Piatetski-Shapiro normality criterion for continued fractions.
Fundamentalnaya i Prikladnaya Matematika, 16:6 (2010) 177–188
- On an inverse theorem for U³-norm.
Modern Problems of Mathematics and Mechanics (2009) 55–127
- On sumsets of dissociated sets. Online Journal of Analytic Combinatorics, 4 (2009) 1–26
- On a two-dimensional analog of Szemerédi’s theorem in abelian groups.
Izvestiya of the Russian Academy of Sciences, 73:5 (2009) 455–505
- On sets with small doubling. Mathematical Notes, 84:6 (2008) 927–947
- On sets of large exponential sums. Izvestiya of the Russian Academy of Sciences, 72:1 (2008) 161–182
- On multiplicative properties modulo m of numbers with missing digits. (with N.G. Moshchevitin) Mathematical Notes, 81:3 (2007) 385–404
- Some examples of sets of large exponential sums. Mathematical Sbornik, 198:12 (2007) 105–140
- On sets of large exponential sums. Doklady of the Russian Academy of Sciences, 411:4 (2006) 455–459
- Szemerédi’s theorem and problems concerning arithmetic progressions.
Russian Mathematical Surveys, 61:6 (2006) 1101–1166
- On a generalization of Szemerédi’s theorem.
Proceedings of the London Mathematical Society, 93:3 (2006) 723–760
- On one problem of Gowers. Izvestiya of the Russian Academy of Sciences, 70:2 (2006) 176–217
- On a generalization of Szemerédi’s theorem. Doklady of the Russian Academy of Sciences, 405:3 (2005) 315–319
- On one problem of Gowers. Doklady of the Russian Academy of Sciences, 400:2 (2005) 169–172
- On the Piatetski-Shapiro normality criterion. Mathematical Notes, 73:4 (2003) 577–589
- Some additive problems connected with exponential functions. Russian Mathematical Surveys, 58:4 (2003) 165–166
- Recurrence of incomplete quotients of continued fractions. Russian Mathematical Surveys, 57:4 (2002) 189–190
- Recurrence in mean.
Mathematical Notes, 72:4 (2002) 625–632
On Fourier coefficients of sets with small doubling.
E–JC (accepted).
On universal sets and sumsets.
Israel Journal of Mathematics (accepted).
Uncertainty for convolutions of sets.
Proc. AMS, 154:2 (2026) 615–627; DOI:10.1090/proc/17485
Some new results on the higher energies.
JNT, 281 (2026) 110–138; DOI:10.1016/j.jnt.2025.09.018
The popularity gap. (with V. Lev)
Journal of Algebraic Combinatorics, 58 (2023) 1155–1172; DOI:10.1007/s10801-023-01268-w
On an application of higher energies to Sidon sets.
Combinatorica, 43 (2023) 329–345; DOI:10.1007/s00493-023-00013-y
Small doubling in prime-order groups: from 2.4 to 2.6. (with V.F. Lev)
Journal of Number Theory, 217 (2020) 278–291; DOI:10.1016/j.jnt.2020.05.009
Double character sums with intervals and arbitrary sets in finite fields. (with I.E. Shparlinski)
Proceedings of the Steklov Institute of Mathematics, 303 (2018) 1–21
Structural theorems in additive combinatorics.
Russian Mathematical Surveys, 70:1 (2015) 123–178
Energies and structure of additive sets.
Electronic Journal of Combinatorics, 21:3 (2014) P3.44
On exponential sums over multiplicative subgroups of medium size.
Finite Fields and Their Applications, 30 (2014) 72–87
Some new results on higher energies.
Transactions of Moscow Mathematical Society, 74:1 (2013) 35–73
On Heilbronn’s exponential sum.
Quarterly Journal of Mathematics, 64:4 (2013) 1221–1230
Some new inequalities in additive combinatorics.
Moscow Journal of Combinatorics and Number Theory, 3:2 (2013) 237–288
Higher moments of convolutions. (with T. Schoen)
Journal of Number Theory, 133 (2013) 1693–1737
Additive properties of multiplicative subgroups of F_p. (with T. Schoen)
Quarterly Journal of Mathematics, 63:3 (2012) 713–722
Some applications of W. Rudin’s inequality to combinatorial number theory.
Uniform Distribution Theory, 6:2 (2011) 95–116
On a paucity result in Incidence Geometry.
Forum Mathematicum (accepted).
The sum-product problem for integers with few prime factors. (with B. Hanson, M. Rudnev and D. Zhelezov)
Compositio Mathematica, 161:3 (2025) 427–446; DOI:10.1112/S0010437X24007735
On the determinants of matrices with elements from arbitrary sets. (with I.E. Shparlinski)
Mathematika, 71:2 (2025) 1–18; DOI:10.1112/mtk.70018
On some multiplicative properties of large difference sets.
Canadian Journal of Mathematics, 76:5 (2024) 1538–1555; DOI:10.4153/S0008414X23000500
On multiplicative Chung–Diaconis–Graham process.
Mat. Sbornik, 214:6 (2023) 136–154; DOI:10.4213/sm9811
Character sums estimates and an application to a problem of Balog. (with T. Schoen)
Indiana University Mathematics Journal, 71:3 (2022) 953–964
On multiplicative energy of subsets of varieties.
Canadian Journal of Mathematics (2022) 1–19; DOI:10.4153/S0008414X21000687
On sums and products of combinatorial cubes.
Finite Fields and Their Applications, 77 (2022) 101948; DOI:10.1016/j.ffa.2021.101948
Bounds of trilinear and trinomial exponential sums. (with S. Macourt, G. Petridis and I.E. Shparlinski)
SIAM Journal on Discrete Mathematics, 34:4 (2020) 2124–2136; DOI:10.1137/20M1325502
On popular sums and differences of sets with small products. (with K.I. Olmezov and A.S. Semchankau)
Mathematical Notes, 108:4 (2020) 561–571
Stronger sum-product inequalities for small sets. (with G. Shakan and M. Rudnev)
Proceedings of the American Mathematical Society, 148 (2020) 1467–1479; DOI:10.1090/proc/14902
Any small multiplicative subgroup is not a sumset.
Finite Fields and Their Applications, 63 (2020) 101645; DOI:10.1016/j.ffa.2020.101645
On the energy variant of the sum-product conjecture. (with M. Rudnev and S. Stevens)
Revista Matemática Iberoamericana, 36:1 (2020) 207–232; DOI:10.4171/RMI/1126
On the few products, many sums problem. (with B. Murphy, M. Rudnev and Y.N. Shteinikov)
Journal de Théorie des Nombres de Bordeaux, 31:3 (2019) 573–602
Some remarks on products of sets in the Heisenberg group and in the affine group.
Forum Mathematicum, 32:1 (2019) 189–199; DOI:10.1515/forum-2019-0181
New results on sum-product type growth over fields. (with B. Murphy, G. Petridis, O. Roche-Newton and M. Rudnev)
Mathematika, 65:3 (2019) 588–642
Some remarks on the asymmetric sum-product phenomenon.
Moscow Journal of Combinatorics and Number Theory, 8:1 (2019) 15–41; DOI:10.2140/moscow.2019.8.15
If A+A is small then AAA is superquadratic. (with O. Roche-Newton) Journal of Number Theory, 201 (2019) 124–134 On the size of the set AA+A. (with O. Roche-Newton, I.Z. Ruzsa and C.-Y. Shen)
Journal of the London Mathematical Society, 99:2 (2019) 477–494
A note on the set A(A+A). (with P.-Y. Bienvenu and F. Hennecart)
Moscow Journal of Combinatorics and Number Theory, 8:2 (2019) 179–188; DOI:10.2140/moscow.2019.8.179
On the restriction problem for discrete paraboloid in lower dimension. (with M. Rudnev)
Advances in Mathematics, 339 (2018) 657–671; DOI:10.1016/j.aim.2018.10.002
Multiplicative Energy of Shifted Subgroups and Bounds on Exponential Sums with Trinomials in Finite Fields. (with S. Macourt and I.E. Shparlinski)
Canadian Journal of Mathematics, 70:6 (2018) 1319–1338; DOI:10.4153/CJM-2017-044-2
On asymptotic formulae in some sum-product questions.
Transactions of the Moscow Mathematical Society, 79:2 (2018) 271–334
On additive bases of sets with small product set. (with D. Zhelezov)
International Mathematics Research Notices, 2018:5 (2018) 1585–1599; DOI:10.1093/imrn/rnw291
Differences of subgroups in subgroups.
International Journal of Number Theory, 14:4 (2018) 1111–1134; DOI:10.1142/S1793042118500690
An application of the sum-product phenomenon to sets having no solutions of several linear equations.
Sbornik: Mathematics, 209:4 (2018) 117–142; DOI:10.1070/SM8907
Variations on the sum-product problem II. (with B. Murphy and O. Roche-Newton)
SIAM Journal on Discrete Mathematics, 31:3 (2017) 1878–1894
On some multiple character sums. (with I.E. Shparlinski)
Mathematika, 63:2 (2017) 553–560; DOI:10.1112/S0025579317000055
Sums of multiplicative characters with additive convolutions. (with A.S. Volostnov)
Proceedings of the Steklov Institute of Mathematics, 296 (2017) 265–279
Some remarks on the Balog–Wooley decomposition theorem and quantities D⁺, D×. Proceedings of the Steklov Institute of Mathematics, 298:1 (2017) 74–90 Growth Estimates in Positive Characteristic via Collisions. (with E. Aksoy Yazici, B. Murphy and M. Rudnev) International Mathematics Research Notices, 2017:23 (2017) 7148–7189; DOI:10.1093/imrn/rnw206 On the number of unit-area triangles spanned by convex grids in the plane. (with O.E. Raz and M. Sharir) Computational Geometry: Theory and Applications, 62 (2017) 25–33; DOI:10.1016/j.comgeo.2016.12.002 On the additive energy of Heilbronn's subgroup. (with E. Solodkova and I. Vyugin) Mathematical Notes, 101:1 (2017) 43–57 Some remarks on sets with small quotient set. Sbornik: Mathematics, 208:12 (2017) 144–158; DOI:10.1070/SM8733 New results on sum-products in R. (with S.V. Konyagin) Proceedings of the Steklov Institute of Mathematics, 294 (2016) 87–98 Difference sets are not multiplicatively closed.
Discrete Analysis, 17 (2016) 1–21; DOI:10.19086/da.913
Intersections of Shifts of Multiplicative Subgroups. (with E. Solodkova and I. Vyugin)
Mathematical Notes, 100:2 (2016) 189–198
On tripling constant of multiplicative subgroups.
INTEGERS, 16 (2016) A75
New sum-product type estimates over finite fields. (with O. Roche-Newton and M. Rudnev)
Advances in Mathematics, 293 (2016) 589–605
Variations on the sum-product problem. (with B. Murphy and O. Roche-Newton)
SIAM Journal on Discrete Mathematics, 29:1 (2015) 514–540
On sum sets of sets having small product sets. (with S.V. Konyagin)
Proceedings of the Steklov Institute of Mathematics, 290 (2015) 304–316
On a question of A. Balog.
Pacific Journal of Mathematics, 280:1 (2015) 227–240
On sums of Szemerédi–Trotter sets.
Proceedings of the Steklov Institute of Mathematics, 289 (2015) 300–309
On additive shifts of multiplicative subgroups. (with I.V. Vyugin)
Mathematical Sbornik, 203:6 (2012) 81–100
On sumsets of convex sets. (with T. Schoen)
Combinatorics, Probability and Computing, 20 (2011) 793–798
On the exceptional set in Littlewood's discrete conjecture.
Bull. LMS, 58:5 (2026) 1-10; DOI:10.1112/blms.70359
On some results of Rogers.
CNT (submitted).
On some results of Korobov and Larcher and Zaremba's conjecture.
Submitted.
Some applications of representation theory to the sum-product phenomenon.
Submitted.
On Korobov bound concerning Zaremba's conjecture. (with N.G. Moshchevitin and B. Murphy)
IMRN, Volume 2026, Issue 6 (2026); DOI:10.1093/imrn/rnag048
On a girth-free variant of the Bourgain–Gamburd machine.
Finite Fields and Their Applications, 90 (2023) 102225; DOI:10.1016/j.ffa.2023.102225
Growth in Chevalley groups relatively to parabolic subgroups and some applications.
Revista Matemática Iberoamericana, 38:6 (2022) 1945–1973; DOI:10.4171/RMI/1344
On growth rate in SL₂(Fp), the affine group and sum-product type implications. (with M. Rudnev)
Mathematika, 68:3 (2022) 738–783; DOI:10.1112/mtk.12120
Noncommutative methods in Additive Combinatorics and Number Theory.
Uspekhi Mat. Nauk, 76:6 (462) (2021) 119–180
Modular hyperbolas and bilinear forms of Kloosterman sums.
Journal of Number Theory, 220 (2021) 182–211; DOI:10.1016/j.jnt.2020.06.014
On a modular form of Zaremba's conjecture. (with N.G. Moshchevitin)
Pacific Journal of Mathematics, 309:1 (2020) 195–211; DOI:10.2140/pjm.2020.309.195
Popular Products and Continued Fractions. (with B. Murphy and N.G. Moshchevitin)
Israel Journal of Mathematics, 238 (2020) 807–835; DOI:10.1007/s11856-020-2039-3
Beyond Threshold Security: Additive Secret Sharing under Hamming-Weight Leakage.
(with A. Biswas, J. Hwang, H.K. Maji and X. Ye)
67th IEEE Symposium On Foundations Of Computer Science (submitted).
On the random Chowla conjecture. (with O. Klurman and M. Xu)
GAFA, 33 (2023) 749–777; DOI:10.1007/s00039-023-00641-y
Number of A+B ≠ C solutions in abelian groups and application to counting independent sets in hypergraphs. (with A. Semchankau and D. Shabanov)
European Journal of Combinatorics, 100 (2022) 103453; DOI:10.1016/j.ejc.2021.103453
On some applications of GCD sums to Arithmetic Combinatorics.
Journal of Number Theory, 231 (2022) 433–450; DOI:10.1016/j.jnt.2021.05.011
Sets of transfer times with small densities. (with M. Björklund and A. Fish)
Journal de l'École polytechnique—Mathématiques, 8 (2021) 311–329; DOI:10.5802/jep.147
Quantitative version of Beurling–Helson theorem. (with S.V. Konyagin)
Functional Analysis and Its Applications, 49:2 (2015) 39–53
On some problems of Euclidean Ramsey theory.
Analysis Mathematica, 41 (2015) 1–12; DOI:10.1007/s10476-015
Kolmogorov width of discrete linear spaces: an approach to matrix rigidity. (with A. Samorodnitsky and S. Yekhanin)
CCC (2015) 1–18
On Gowers norms of some functions.
Mathematical Notes, 92:4 (2012) 609–627
On dynamical systems with slow recurrence time.
Mathematical Sbornik, 197:11 (2006) 143–158
Office: MATH 736