Explicit constructions of extractors and expanders

@article{Hegyvari2009ExplicitCO,
  title={Explicit constructions of extractors and expanders},
  author={Norbert Hegyv'ari and Françcois Hennecart},
  journal={Acta Arithmetica},
  year={2009},
  volume={140},
  pages={233-249}
}
We investigate 2-variable expanders and 3-source extractors in prime fields. We extend previous results of J. Bourgain. 
Some remarks on multilinear exponential sums with an application
Moderate expanders over rings
Erdős–Rényi graph, Szemerédi–Trotter type theorem, and sum-product estimates over finite rings
TLDR
A variant of Erdős–Rényi graph over finite rings is studied and a Szemerédi–Trotter type theorem is obtained a sum-product estimate in finite rings.
Distribution of residues in approximate subgroups of F*p
We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type f(I) ·H is equidistributed (as p tends to infinity) where f is a polynomial, I is an
Sums and products of sets and estimates of rational trigonometric sums in fields of prime order
This paper is a survey of main results on the problem of sums and products of sets in fields of prime order and their applications to estimates of rational trigonometric sums. Bibliography: 85 titles.
On sum-product bases
Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that
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
On the cone restriction conjecture in four dimensions and applications in incidence geometry
The first purpose of this paper is to solve the cone restriction conjecture in four dimensions in the finite field setting. We will also clarify conjectures for higher dimensional cases. The second
Bounded Collusion Protocols, Cylinder-Intersection Extractors and Leakage-Resilient Secret Sharing
TLDR
This work investigates BCPs more thoroughly and relies on a new class of exponential sums that interpolate the ones in additive combinatorics by Bourgain (Geometric and Functional and Petridis and Shparlinski).
Fourier analysis in combinatorial number theory
In this survey applications of harmonic analysis to combinatorial number theory are considered. Discussion topics include classical problems of additive combinatorics, colouring problems,
...
...

References

SHOWING 1-10 OF 12 REFERENCES
On Exponential Sums in Finite Fields
The purpose of this paper is to establish certain multilinear exponential sums in arbitrary finite fields, extending some of the results from [1] for prime fields.
The Szemerédi-Trotter type theorem and the sum-product estimate in finite fields
  • L. Vinh
  • Mathematics
    Eur. J. Comb.
  • 2011
Some remarks on number theory. II.
Like the previous paper of the same title [5] this note contains disconnected remarks on number theory.
MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS
In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when
Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors
TLDR
The following new explicit instructions are given for the construction of deterministic extractors, dispersers and related objects for any fixed rate δ‹1⁄2 and no previous explicit construction was known for either of these.
SOME REMARKS ON NUMBER THEORY
This note contains some disconnected minor remarks on number theory . 1 . Let (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle . Put n s(k, n) _ z~, Ak = Jim sup I s(k, n)
A sum-product theorem in finite fields, and applications, Geom
  • Funct. Anal
  • 2004
On sums and products of residues modulo p
An asymptotic inequality in the theory of numbers
  • Vestnik Leningrad Univ .
...
...