MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

By Jean Bourgain. 2005

314 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/195e74c550fa102fdcbe327dfc70183f8aefc4db

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In Artificial Intelligence

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Modular hyperbolas (2011, background): “Recently, an explicit version of the results of [31] has been given in [14] how it is not immediately clear whether it can lead to new results about the congruence (21) (except for the case where the variables are in the domain p1 6 X , p2, p3 6 Y with X substantially smaller than Y ).”
Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications (2010, methodology): “Using Bourgain’s sum-product estimate [1], they have shown that there exists an effectively computable constant η > 0 such that for any positive integer M < p, uniformly over arbitrary integers K and L, the following bound holds: I2(M ; K, L) ¿ M 2 p + M1−η.”
THE SUM-PRODUCT ESTIMATE FOR LARGE SUBSETS OF PRIME FIELDS (2007, background): “The motivation for the quantity |A| 1/2p1/2−ε is clear from the construction in [3] which can be described as follows.”
2-source dispersers for $n^{o(1)}$ entropy, and Ramsey graphs beating the Frankl-Wilson construction (2012, methodology): “In particular, we rely on the extractors of Raz [Raz05] and Bourgain [Bou05] as well as an improved version of the extractor of Rao [Rao06].”
Growth Estimates in Positive Characteristic via Collisions (2015, methodology): “, by Bourgain [4] and Bukh and Tsimerman [7], see also the references contained therein.”
On the size of the set A(A + 1) (2008, background): “Nevertheless, from Bourgain’s work [ 1 ] it is known that if |A| ∼ p�, where 0 < α < 1,”
On congruences with products of variables from short intervals and applications (2012, methodology): “However, this approach does not give any nontrivial estimates for small values of h, and thus Chan and Shparlinski [5], for ν = 2, have employed methods of additive combinatorics, namely some results of Bourgain [3], in order to obtain a nontrivial upper bound on Jν(p, h, s;λ) for any h. Cilleruelo ”
Algebraic trace functions over the primes (2012, background): “For instance, Bourgain [3] and Bourgain-Garaev [5] have obtained bounds for K(n) = e (an+ bnk p ) , k ∈ N− {0}, (b, p) = 1, which are non-trivial as long as X > p1/2+ε (see also [17] for a survey of existing methods).”
Rainbow Connections of Graphs: A Survey (2011, background): “Their proof employed a very recent breakthrough due to Bourgain [8, 51], which consists of a powerful explicit extractor.”
On asymptotic formulae in some sum–product questions (2018, background): “Some applications of such sort of bounds can be found in papers [4], [50].”
An improved point‐line incidence bound over arbitrary fields (2016, background): “This improves the exponent for a result of Bourgain [6].”
On the Construction of Affine Extractors (2007, methodology): “Let us point out the following consequence of Theorem 0.2 and Proposition 0.3 (see also [ Bo ] for related results).”
Quantum random number generators (2016, methodology): “Other representative methods to combine different randomness sources can be found in (Barak et al., 2006; Bourgain, 2005; Dodis et al., 2004; Rao, 2009; Raz, 2005; Shaltiel, 2008).”
Sum–product estimates for rational functions (2010, methodology): “In application to a construction of extractors, in the same paper Bourgain asked for sum–product estimates for f(x1, . . . , xk) = xt1 + · · · + xtk. Received 12 February 2010; revised 15 March 2011; published online 4 August 2011.”
Linear degree extractors and the inapproximability of max clique and chromatic number (2006, unknown): “Second, we strengthen a theorem of Bourgain [10] and show that the function A(A + B) also gives a rate improvement.”