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. 
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References

SHOWING 1-10 OF 50 REFERENCES
On the Number of Incidences Between Points and Curves
We apply an idea of Szekely to prove a general upper bound on the number of incidences between a set of m points and a set of n ‘well-behaved’ curves in the plane.
Incidences with Curves in R
We prove that the number of incidences between m points and n bounded-degree curves with k degrees of freedom in R is
The Szemerédi-Trotter type theorem and the sum-product estimate in finite fields
  • L. Vinh
  • Mathematics, Computer Science
  • Eur. J. Comb.
  • 2011
TLDR
A Szemeredi-Trotter type theorem is studied and used to obtain a different proof of Garaev's sum-product estimate in finite fields. Expand
A Kakeya-type problem for circles
We prove full Hausdorff dimension in a variant of the Kakeya problem involving circles in the plane, and also sharp estimates for the relevant maximal function. These results can also be formulatedExpand
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
  • L. Székely
  • Mathematics, Computer Science
  • Comb. Probab. Comput.
  • 1997
We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: theExpand
The discretized sum-product and projection theorems
We give a new presentation of the discrete ring theorem for sets of real numbers [B]. Special attention is given to the relation between the various parameters. As an application, new Marstrand typeExpand
Weighted restriction estimates and application to Falconer distance set problem
abstract:We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of theExpand
Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type
In this paper we investigate three unsolved conjectures in geomet- ric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. WeExpand
The proof of the $l^2$ Decoupling Conjecture
We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of themExpand
An improved point‐line incidence bound over arbitrary fields
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show thatExpand
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