# Incidence Estimates for Well Spaced Tubes

@article{Guth2019IncidenceEF, title={Incidence Estimates for Well Spaced Tubes}, author={L. Guth and Noam Solomon and H. Wang}, journal={Geometric and Functional Analysis}, year={2019} }

We prove analogues of the Szemeredi-Trotter theorem and other incidence theorems using $\delta$-tubes in place of straight lines, assuming that the $\delta$-tubes are well-spaced in a strong sense.

#### 13 Citations

An incidence estimate and a Furstenberg type estimate for tubes in $\mathbb{R}^2$

- Mathematics
- 2021

We study the δ-discretized Szemerédi-Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both two problems,… Expand

An Incidence Result for Well-Spaced Atoms in all Dimensions

- Mathematics
- 2020

We prove an incidence result counting the $k$-rich $\delta$-tubes induced by a well-spaced set of $\delta$-atoms. Our result coincides with the bound that would be heuristically predicted by the… Expand

INCIDENCE ESTIMATES FOR α-DIMENSIONAL TUBES

- 2020

Given a collection of α-dimensional δ-tubes and β-dimensional δballs in the plane, we consider the problem of finding the maximum number of incidences, or pairs (t, b) of tubes and balls such that b… Expand

A sharp $L^{10}$ decoupling for the twisted cubic

- Mathematics
- 2020

We prove a sharp $l^{10}(L^{10})$ decoupling for the moment curve in $\mathbb{R}^3$. The proof involves a two-step decoupling combined with new incidence estimates for planks, tubes and plates.

A DECOUPLING INEQUALITY FOR SHORT GENERALIZED DIRICHLET SEQUENCES

- 2021

We prove an lL decoupling inequality for functions on R with Fourier transform supported in a neighborhood of short Dirichlet sequences {log n} 1/2 n=N+1 , as well as sequences with similar convexity… Expand

Decoupling inequalities for short generalized Dirichlet sequences

- Mathematics
- 2021

We study decoupling theory for functions on R with Fourier transform supported in a neighborhood of short Dirichlet sequences {log n} 1/2 n=N+1 , as well as sequences with similar convexity… Expand

A nonlinear version of Bourgain's projection theorem

- Mathematics
- 2020

We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a… Expand

A Marstrand type slicing theorem for subsets of $\mathbb{Z}^2 \subset \mathbb{R}^2$ with the mass dimension

- Mathematics
- 2020

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima… Expand

A Sum-Product Estimate for Well Spaced Sets

- Mathematics
- 2020

We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set… Expand

Planar incidences and geometric inequalities in the Heisenberg group

- Mathematics
- 2020

We prove that if $P,\mathcal{L}$ are finite sets of $\delta$-separated points and lines in $\mathbb{R}^{2}$, the number of $\delta$-incidences between $P$ and $\mathcal{L}$ is no larger than a… Expand

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