From harmonic analysis to arithmetic combinatorics

Łaba, Izabella

doi:10.1090/s0273-0979-07-01189-5

Cited by 23 publications

(18 citation statements)

References 181 publications

(254 reference statements)

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“…The Nikodym conjecture is a close cousin of the more well-known Kakeya conjecture. The connection between these geometrical questions and problems in Fourier analysis and PDE is described in [Laba08] and [Tao01].…”

Section: Examples Of the Polynomial Methodsmentioning

confidence: 99%

Unexpected Applications of Polynomials in Combinatorics

Guth

2013

The Mathematics of Paul Erdős I

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In the last six years, several combinatorics problems have been solved in an unexpected way using high degree polynomials. The most well-known of these problems is the distinct distance problem in the plane. In [Erdős46], Erdős asked what is the smallest number of distinct distances determined by n points in the plane. He noted that a square grid determines ∼ n(log n) −1/2 distinct distances, and he conjectured that this is sharp up to constant factors. Recently, an estimate was proven which is sharp up to logarithmic factors. The main new thing in the proof is the use of high-degree polynomials. This new technique first appeared in Dvir's paper [Dvir09], which solved the finite field Nikodym and Kakeya problems. Experts had considered these problems very difficult, but the proof was essentially one page long. The method has had several other applications. The joints problem was resolved in . The argument was simplified and generalized in [KSS10] and [Quilodrán10], leading to another one page proof. A higher-dimensional generalization of the Szemerédi-Trotter theorem was proven in . And several fundamental theorems in incidence geometry were reproved in the paper [KMS12].The new trick in these proofs can be summarized as follows. We want to understand some finite set S in a vector space. We consider a minimal degree (non-zero) polynomial that vanishes on the set S. Then we use this polynomial to study the problem. This strategy is somewhat surprising because the statements of the problems often involve only points and lines. The joints problem and the finite field Nikodym problem can be solved in a page each using high degree polynomials but seem very difficult to solve without polynomials. Why polynomials play such a crucial role in these problems is somewhat mysterious.The point of this essay is to explain how these new methods work and to reflect on them philosophically. The main theme is the connection between combinatorics and algebra (polynomials).Here is an outline of the essay. We begin by giving two detailed examples of the polynomial method: the finite field Nikodym problem and the joints problem. This is the subject of Section 1: Examples of the polynomial method.Once we've seen a couple examples of this method, we're going to work on understanding "where it comes from". In Section 2, we discuss where the method comes from historically.

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Section: Examples Of the Polynomial Methodsmentioning

confidence: 99%

Unexpected Applications of Polynomials in Combinatorics

Guth

2013

The Mathematics of Paul Erdős I

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“…What happens X ·X contains unusually few elements; for example what if |X · X| ≤ 2|X|? The structure of such sets is studied in additive combinatorics, which is a burgeoning area of mathematics with applications in computer science [35], harmonic analysis [25], number theory [28], combinatorics and elsewhere. It has spanned a host of new techniques (e.g.…”

Section: 2mentioning

confidence: 99%

Carries, Group Theory, and Additive Combinatorics

Diaconis

Shao²,

Soundararajan³

2014

The American Mathematical Monthly

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“…The structure of incidences between points and various geometric objects is of central importance in discrete geometry, and theorems that elucidate this structure have had applications to, for example, problems from discrete and computational geometry [18,23], additive combinatorics [8,13], harmonic analysis [17,22], and computer science [12]. The study of incidence theorems for finite geometry is an active area of researche.g.…”

Section: Introductionmentioning

confidence: 99%

Incidence Bounds for Block Designs

Lund

Saraf²

2016

SIAM J. Discrete Math.

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We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of incidences between points and m-flats in affine geometries over finite fields. First, we show an upper bound on the number of incidences between sufficiently large subsets of the points and blocks of a BIBD. Second, we show that a sufficiently large subset of the points of a BIBD determines many t-rich blocks. Third, we show that a sufficiently large subset of the blocks of a BIBD determines many t-rich points. These last two results are new even in the special case of incidences between points and m-flats in an affine geometry over a finite field.As a corollary we obtain a tight bound on the number of t-rich points determined by a set of points in a plane over a finite field, and use it to sharpen a result of Iosevich, Rudnev, and Zhai [20] on the number of triangles with distinct areas determined by a set of points in a plane over a finite field.

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From harmonic analysis to arithmetic combinatorics

Cited by 23 publications

References 181 publications

Unexpected Applications of Polynomials in Combinatorics

Unexpected Applications of Polynomials in Combinatorics

Carries, Group Theory, and Additive Combinatorics

Incidence Bounds for Block Designs

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