The discretized sum-product and projection theorems

By Jean Bourgain. 2010

142 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b0ca5b065de228723b8030622477e8d21622c1d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On self-similar sets with overlaps and inverse theorems for entropy (2012, methodology): “The study of parametric families of general sets and measures is classical; examples include the projection theorems of Besicovitch and Marstrand and more recent results like those of Peres-Schlag [24] and Bourgain [3].”
An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³ (2017, background): “A precise version of this statement is needed in order to apply Bourgain’s discretized sum-product theorem.”
Recent progress on Bernoulli convolutions (2016, methodology): “This result can be thought of as an entropy analogue of the additive part in the proof of Bourgain’s discretized sum product theorem [4], [5].”
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane (2021, background): “As mentioned already, it follows by combining results of Katz-Tao [10] and Bourgain [1] that s-Furstenberg sets have Hausdorff dimension at least 1 ` for s "very close" to 1 2 .”
A Spectral Gap Theorem in $SU(d)$ (2011, background): “urely combinatorial”, culminating in Propositions 2, 6 and 7 that are needed for the SU(d) 4 analysis. The ﬁrst part is closely related to the proof of the discretized ring theorem in R, presented in [B2]; the generalization to the higher dimensional setting necessitates reproducing several technica”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “This fact goes back at least to [5]; we provide details for completeness.”
Absolute continuity of Bernoulli convolutions for algebraic parameters (2016, background): “This is in contrast with [13] (and [2], where the size of sum-sets is estimated instead of the entropy of convolutions), where the entropy of the convolution product of a large number of measures is estimated first and then Plünnecke-type inequalities are used.”
New Bounds on the Dimensions of Planar Distance Sets (2018, methodology): “eorems is to decompose an arbitrary 2 T‘-measure in terms of measures which have a uniform tree structure when represented in base 2T. This notion (which is inspired by some constructions of Bourgain [3]) is made precise in the next deﬁnition. Deﬁnition 3.2. Given a sequence ˙= (˙ 1;:::;˙ ‘) 2R‘, we”
Zaremba’s conjecture (2011, background): “ the problem of traces. We plan to return to this problem in the future. The proof of Theorem1.22follows along completely dierent lines, and is inspired by the recent advances in projection theorems [Bou10]. The observation here is that the set D has a certain \sum-set" structure. Namely, if b=”
Fourier dimension and spectral gaps for hyperbolic surfaces (2017, methodology): “atterson{ Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group are nonlinear, together with estimates on exponential sums due to Bourgain [Bou10] which follow from the discretized sum-product theorem in R. Let M= nH2 be a (noncompact) con”