My research lies mainly in the areas of classical harmonic analysis and discrete geometry. In particular, I study questions relating to Kakeya-type maximal functions and questions in incidence geometry such as the unit and distinct distance problems, and point-curve or point-surface incidence bounds.

Papers and Preprints

[By Area]   [By year]

2024

• The Assouad dimension of Kakeya sets in ℝ³ (with H. Wang)

  Submitted (2024) [arXiv].

2023

• On Maximal Functions Associated to Families of Curves in the Plane.

  Submitted (2023) [arXiv].

2022

• Improved Elekes-Szabó type estimates using proximity (with J. Solymosi

  J. Comb. Theory Ser. A. (2024) 201:105813. [arXiv] [Journal]

• Kakeya sets from lines in SL₂ (with N.H. Katz and S. Wu

  Ars Inven. Anal. (2023) Paper No. 6, 23 pp. [arXiv] [Journal].

• Sticky Kakeya sets and the sticky Kakeya conjecture (with H. Wang)

  Submitted (2022) [arXiv].

• Unions of lines in ℝⁿ

  Mathematika (2023) 69(2):473--481. [arXiv] [Journal].

• A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in ℝ³ (with M. Pramanik, T. Yang)

  accepted, Amer. J. Math.. [arXiv]

2021

• New estimates on the size of (α,2α)-Furstenberg sets (with D. Di Benedetto)

  preprint. [arXiv]

• On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem (with O. Raz)

  Geom. Funct. Anal. (GAFA) (2024) 34, 209–262. [arXiv] [Journal]

2020

• A note on Fourier restriction and nested Polynomial Wolff axioms (with J. Hickman)

  J. Anal. Math. (2024) 152:19--52.[arXiv] [Journal]

• Counting and cutting rich lenses in arrangements of circles (with E. Ezra, O. Raz, M. Sharir

  SIAM J. Discrete Math. (2022) 36(2): 958–974.[arXiv] [Journal]
  Preliminary version in Proc. 37th International Symposium on Computational Geometry (SoCG 2021) [Conference Proceedings]

• Distinct distances in the complex plane (with A. Sheffer

  Trans. Amer. Math. Soc. (2021) 374(9): 6691–6725. [arXiv] [Journal]

• Sphere tangencies, line incidences, and Lie's line-sphere correspondence

  Math. Proc. Camb. Philos. Soc. (2022) 172:2, 401–421, [arXiv] [Journal]

2019

• New Kakeya estimates using Gromov's algebraic lemma

  Adv. Math. (2021): 380 [arXiv] [Journal]

• A Kakeya maximal function estimate in four dimensions using planebrushes (with N.H. Katz)

  Rev. Mat. Iberoam. (2021) 37(1):317--359. [arXiv] [Journal]

• Large Sets Avoiding Rough Patterns (with J. Denson, M. Pramanik)

  In: Rassias M.T. (eds) Harmonic Analysis and Applications, pp 59-75. Springer Optimization and Its Applications, vol 168. Springer, 2021. [arXiv] [Book chapter]

2018

• A discretized Severi-type theorem with applications to harmonic analysis

  Geom. Funct. Anal. (GAFA) (2018) 28(4):1131--1181. [arXiv] [Journal]

• On the discretized sum-product problem (with L. Guth, N.H. Katz)

  Int. Math. Res. Not. (IMRN) (2021) Volume 2021, Issue 13: 9769–9785. [arXiv] [Journal]

• Counting higher order tangencies for plane curves

  Combin. Probab. Comput. (2020) 29(2): 310–317. [arXiv] [Journal]

• An efficient algorithm for generalized polynomial partitioning and its applications (with P. Agarwal, B. Aronov, E. Ezra)

  SIAM J. Comput. (2021) 50(2):760–787. [arXiv] [Journal].
  Preliminary version in Proc. 35th International Symposium on Computational Geometry (SoCG 2019) [Conference Proc.].

• Constructive polynomial partitioning for algebraic curves in ℝ³ with applications (with B. Aronov, E. Ezra)

  SIAM J. Comput. (2020) 49(6):1109–1127. [arXiv][Journal]
  Preliminary version in Proc. Thirtieth Annu. ACM-SIAM Sympos. Discr. Alg. 2636-2648, 2019. [Conference Proc.]

2017

• An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³ (with N.H. Katz)

  J. Amer. Math. Soc. (2019) 32(1):195--259. [arXiv] [Journal]

• Polynomial Wolff axioms and Kakeya-type estimates in ℝ⁴ (with L. Guth)

  Proc. London Math. Soc. (2018) 117(1): 192--220. [arXiv] [Journal]

• Breaking the 3/2 barrier for unit distances in three dimensions

  Int. Math. Res. Not. (IMRN) (2019) Vol 2019, Issue 20: 6235–6284.[arXiv] [Journal] [Erratum]

2016

• Cutting algebraic curves into pseudo-segments and applications (with M. Sharir)

  J. Combin. Theory Ser. A (2017) 150:1–35, . [arXiv] [Journal]

2015

• Curves in ℝ⁴ and two-rich points (with L.~Guth)

  Discrete. Comput. Geom. (2017) 58(1): 232–253. [arXiv] [Journal]

• Spectral gaps, additive energy, and a fractal uncertainty principle (with S. Dyatlov)

  Geom. Funct. Anal. (GAFA) (2016) 26(4):1011–1094. [arXiv] [Journal]

• New bounds on curve tangencies and orthogonalities (with J. Ellenberg, J. Solymosi)

  Discrete Analysis (2016) 22:1–22. [arXiv] [Journal]

• Algebraic curves, rich points, and doubly-ruled surfaces (with L. Guth)

  Amer. J. Math. (2018) 140(5): 1187-1229. [arXiv] [Journal]

• A note on rich lines in truly high dimensional sets

  FoM, Sigma (2016) 4(e2):1–13. [arXiv] [Journal]

• Point-curve incidences in the complex plane (with A. Sheffer, E. Szabó)

  Combinatorica (2018) 38(2): 487--499. [arXiv] [Journal]

2014

• A semi-algebraic version of Zarankiewicz's problem (with J. Fox, J. Pach, A. Sheffer, A. Suk)

  J. Eur. Math. Soc. (2017) 19(6): 1785–1810. [arXiv] [Journal]

2013

• Few distinct distances implies no heavy lines or circles (with A. Sheffer, F. de Zeeuw)

  Combinatorica (2016) 36(3): 349--364. [arXiv] [Journal]

• Quantitative visibility estimates for unrectifiable sets in the plane (with M. Bond, I. Łaba)

  Trans. Amer. Math. Soc (2016) 368, 5475-5513. [arXiv] [Journal]

2012

• Incidences between points and non-coplanar circles (with A. Sheffer, M. Sharir)

  Combin. Probab. Comput. (2015) 24(3), 490–520.[arXiv] [journal]

• A Szemeredi-Trotter type theorem in ℝ⁴.

  Discrete. Comput. Geom. (2015) 54(3):513–572. [arXiv] [Journal]

2011

• On the Wolff circular maximal function

  Illinois J. Math. (2012) 56(4):1281–1295. [arXiv] [Journal]

• An improved bound on the number of point-surface incidences in three dimensions

  Contrib. Discrete Math. (2013) 8(1):100–121. [arXiv] [Journal] [errata]

2010

• L³ estimates for an algebraic variable coefficient Wolff circular maximal function

  Rev. Mat. Iberoam. (2012) 28(4):1061–1090. [arXiv] [Journal] [Errata]

2007

• Bounds on degrees of p-adic separating polynomials (with D.J. Katz)

  J. Combin. Theory Ser. A (2008) 115(7):1310–1319. [pdf][Journal]

• On universal cycles for multisets (with G. Hurlbert, T. Johnson)

  Discrete Math. (2009) 309(17):5321–5327. [arXiv] [Journal]

Talks

Here are some recordings of talks I have given on my research.

• Lens counting, circular maximal functions, and restricted projections. This was a talk discussing the results of the paper "A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in ℝ³," by myself, Malabika Pramanik and Tongou Yang. It was given at the Oberwolfach Workshop "Real Analysis, Harmonic Analysis, and Applications" on July 7, 2022.
• Dimension-expanding polynomials and the discretized Elekes-Ronyai theorem. This was a talk discussing the results of the paper by the same title, by myself and Orit Raz. It was given at the MSU Mathematics seminar on Feb 22, 2021.
• The discretized sum-product problem This was a talk explaining Bourgain's discretized sum-product theorem and discussing a the results of the paper "On the discretized sum-product problem" by myself, Guth, and Katz. It was given at the NSF-CBMS Conference on Additive Combinatorics from a Geometric Viewpoint on May 25, 2018.
• Unit distances in three dimensions. This was a talk given about some results in the paper "Breaking the 3/2 barrier for unit distances in three dimensions." It was given at the Banff International Research Station on February 5, 2018.
• Cutting curves into segments and incidence geometry. This was a talk given about some results in the papers "Breaking the 3/2 barrier for unit distances in three dimensions" and "Cutting algebraic curves into pseudo-segments and applications." It was given at the Harvard Center of mathematical sciences and applications on November 13, 2017.
• An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³ . This was a talk given about some results in the paper by the same title. It was given at MSRI on May 19, 2017.
• Some questions in discretized additive combinatorics. This was a talk given about some results in the paper "Spectral gaps, additive energy, and a fractal uncertainty principle." It was given at ICERM on February 15, 2016 as part of the workshop "Ergodic, Algebraic and Combinatorial Methods in Dimension Theory."
• Space Curve Arrangements with Many Incidences. This was a talk given about a preliminary version of the results in the paper "Algebraic curves, rich points, and doubly-ruled surfaces." It was given at IPAM on May 22, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Finding Algebraic Structures in Extremal Combinatorial Configurations"
• Visibility and discretized projection theorems . This was a talk given about some results in the paper "Quantitative visibility estimates for unrectifiable sets in the plane." It was given at IPAM on May 8, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: The Kakeya Problem, Restriction Problem, and Sum-product Theory"
• Multi-level partitioning theorems. This was a talk given about some results in the paper "A semi-algebraic version of Zarankiewicz's problem." It was given at IPAM on April 7, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Tools from Algebraic Geometry."
• On the structure of planar point sets that determine few distinct distances. This talk discusses the results in the paper "Few distinct distances implies no heavy lines or circles." It was given at IPAM on March 28, 2014 as part of the workshop "Algebraic Techniques for Combinatorial and Computational Geometry: Combinatorial Geometry Problems at the Algebraic Interface"