Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

Abstract

We show that n arbitrary circles in the plane can be cut into O(n 3/2+ε ) arcs, for any ε > 0, such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.

Geometric incidence theorems via Fourier analysis

“…A model case that the reader should keep in mind is that of a finite and homogeneous point set A ⊂ [0, 1] d and the family of spheres {S a } a∈A , where S a = {x ∈ R d : |x − a| = .1}. We will obtain an upper bound on the number of pairs…”

From harmonic analysis to arithmetic combinatorics

“…Besicovitch's construction shows that Kakeya sets in dimension 2 can have measure 0. With this information, it is easy to deduce a similar result in higher dimensions: if E is a planar Kakeya set of measure 0, then the set E × [0, 1] d−2 in R d is a Kakeya set and has d-dimensional measure 0. It turns out, however, that many problems in analysis call for more detailed information on the size of Kakeya sets in terms of their dimension.…”

“…The best current bound in the case m ≈ n is O(n 1.364 ), due to Aronov and Sharir [1]; for general n and m, the estimate is more complicated and distinguishes between several cases. This bound is weaker than (4.1) -as is to be expected, given that circles have more "degrees of freedom" than lines.…”

“…The proof of this follows the rough outline of Furstenberg's ergodic proof of Szemerédi's theorem, but draws also on ideas of other authors, including Gowers [74], [76] and Host-Kra [98]. An inductive procedure is used to decompose f into random and quasiperiodic parts f U and f ⊥ , where f ⊥ is non-negative and bounded, and f U is unbounded but has a very small U k− 1 Gowers norm. The contribution of the random part to (6.1) is negligible.…”

On the complexity of many faces in arrangements of circles

“…(We note, though, that the best known bound for the complexity of many faces in an arrangement of line segments is slightly weaker [5].) The same has been true for arrangements of circles (except for the tiny 4"^"'^' factor in the leading term), until recently, when Aronov and Sharir [7] obtained an improved bound on the number of incidences between points and circles, showing that this number is Oim^'^ri^'^ + ;"6/ii+3£"9/ii-£ + fn + nX for any e > 0. They raised the question whether a similar bound can be obtained for the complexity of many faces in circle arrangements, which, after the cases of lines, segments, and pseudolines, is the next natural problem instance to be tackled.…”