The Uniformity Conjecture in Additive Combinatorics

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

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Equivalence of polynomial conjectures in additive combinatorics

COMBINATORICA ◽

10.1007/s00493-012-2714-z ◽

2012 ◽

Vol 32 (5) ◽

pp. 607-618 ◽

Cited By ~ 4

Author(s):

Shachar Lovett

Keyword(s):

Additive Combinatorics

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RANDOM MATRICES: THE CIRCULAR LAW

Communications in Contemporary Mathematics ◽

10.1142/s0219199708002788 ◽

2008 ◽

Vol 10 (02) ◽

pp. 261-307 ◽

Cited By ~ 71

Author(s):

TERENCE TAO ◽

VAN VU

Keyword(s):

Strong Convergence ◽

Random Matrices ◽

Random Variable ◽

Singular Value ◽

Additive Combinatorics ◽

Sparse Random Matrices ◽

Circular Law ◽

Bounded Variance ◽

Complex Random Variable ◽

Least Singular Value

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

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