#### 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

#### Kakeya and Restriction

- Zahl. New Kakeya estimates using Gromov's algebraic lemma. 2019. Submitted. [arXiv]
- Katz, Zahl. A Kakeya maximal function estimate in four dimensions
using planebrushes. 2019. Accepted,
*Rev. Mat. Iberoam.*[arXiv] - Zahl. A discretized Severi-type theorem with applications to harmonic analysis. 2018.
*GAFA*28(4):1131--1181, 2018. [arXiv] [Journal] - Katz, Zahl. An improved bound on the Hausdorff dimension of Besicovitch sets in
ℝ
^{3}. 2017.*J. Amer. Math. Soc*32(1):195--259, 2019. [arXiv] [Journal] - Guth, Zahl. Polynomial Wolff axioms and Kakeya-type estimates in ℝ
^{4}. 2017.*Proc. London Math. Soc.*117(1): 192--220, 2018. [arXiv] [Journal] - Zahl. On the Wolff circular maximal function. 2011.
*Illinois J. Math.*56(4):1281–1295, 2012(2014). [arXiv] [Journal] - Zahl.
*L*^{3}estimates for an algebraic variable coefficient Wolff circular maximal function. 2010.*Rev. Mat. Iberoam.*28(4):1061–1090, 2012. [arXiv] [Journal] [Errata]

#### Combinatorial Geometry

- Sheffer, Zahl. Distinct distances in the complex plane. 2020. Submitted. [arXiv]
- Zahl. Sphere tangencies, line incidences, and Lie's line-sphere correspondence. 2020. Submitted. [arXiv]
- Zahl. Counting higher order tangencies for plane curves. 2018.
*Combin. Probab. Comput.*29(2):310–317, 2020 [arXiv] [Journal] - Zahl. Breaking the 3/2 barrier for unit distances in three dimensions. 2017.
*IMRN*Vol 2019, Issue 20: 6235–6284, 2019. [arXiv] [Journal] - Sharir, Zahl. Cutting algebraic curves into pseudo-segments and applications. 2016.
*J. Combin. Theory Ser. A*. 150:1–35, 2017. [arXiv] [Journal] - Guth, Zahl. Curves in ℝ
^{4}and two-rich points. 2015.*Discrete. Comput. Geom.*58(1): 232–253, 2017. [arXiv] [Journal] - Ellenberg, Solymosi, Zahl. New bounds
on curve tangencies and orthogonalities. 2015.
*Discrete Analysis*. 22:1–22, 2016 [arXiv] [Journal] - Guth, Zahl. Algebraic curves, rich points, and doubly-ruled surfaces. 2015.
*Amer. J. Math.*140(5): 1187-1229. [arXiv] [Journal] - Zahl. A note on rich lines in truly high dimensional sets. 2015.
*FoM, Sigma*. 4(e2):1–13, 2016 [arXiv] [Journal] - Sheffer, Szabó, Zahl.
Point-curve incidences in the
complex plane. 2015.
*Combinatorica*38(2): 487--499, 2018. [arXiv] [Journal] - Fox, Pach, Sheffer,
Suk, Zahl. A semi-algebraic version of Zarankiewicz's problem. 2014.
*J. Eur. Math. Soc.*. 19(6): 1785–1810, 2017. [arXiv] [Journal] - Sheffer, Zahl, de Zeeuw. Few
distinct distances
implies no heavy lines or circles. 2013.
*Combinatorica*36(3): 349--364, 2016. [arXiv] [Journal] - Sheffer, Sharir, and Zahl.
Incidences between points and non-coplanar
circles. 2012.
*Combin. Probab. Comput.*24(3), 490–520, 2015.[arXiv] [journal] - Zahl. A Szemeredi-Trotter type theorem in ℝ
^{4}. 2012.*Discrete. Comput. Geom.*54(3):513–572, 2015. [arXiv] [Journal] - Zahl. An improved bound on the number of point-surface incidences in three dimensions. 2011.
*Contrib. Discrete Math.*8(1):100–121, 2013. [arXiv] [Journal] [errata]

#### Additive Combinatorics and Geometric Measure Theory

- Denson, Pramanik, Zahl. Large Sets Avoiding Rough
Patterns. 2019. To appear in
*Harmonic Analysis and Applications*, Springer special volume. [arXiv] - Guth, Katz, Zahl.
On the discretized sum-product problem. 2018. In press,
*IMRN*. [arXiv] [Journal] - Dyatlov, Zahl. Spectral gaps, additive energy, and a fractal uncertainty principle.
2015.
*GAFA.*26(4):1011–1094, 2016. [arXiv] [Journal] - Bond, Łaba, Zahl. Quantitative
visibility estimates for unrectifiable sets in the plane. 2013.
*Trans. Amer. Math. Soc*. 368, 5475-5513, 2016. [arXiv] [Journal]

#### Computational Geometry

- Agarwal, Aronov, Ezra, Zahl. An efficient algorithm for generalized polynomial partitioning and its applications.
2018. Submitted. [arXiv].

Preliminary version in*Proc. 35th International Symposium on Computational Geometry (SoCG 2019)*[Conference Proc.]. - Aronov, Ezra, Zahl. Constructive polynomial partitioning for algebraic curves in ℝ
^{3}with applications. 2018. Accepted,*SIAM J. Comput.*[arXiv]

Preliminary version in*Proc. Thirtieth Annu. ACM-SIAM Sympos. Discr. Alg.*2636-2648, 2019. [Conference Proc.]

#### Misc. Combinatorics

- Hurlbert, Johnson,
Zahl. On universal
cycles for multisets. 2007.
*Discrete Math.*, 309(17):5321–5327, 2009. [arXiv] [Journal] - Katz, Zahl. Bounds on degrees of
*p*-adic separating polynomials. 2007.*J. Combin. Theory Ser. A*. 115(7):1310–1319, 2008. [pdf][Journal]

### Talks

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

- 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 ℝ
^{3}. 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"