Borel subrings of the reals

By Gerald A. Edgar, C. Miller. 2002

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Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order (2006, background): “For the proof, following [10] and [5], we associate with a set A ⊂ F the set I(A) := {a1(a2 − a3) + a4(a5 − a6) : a1, .”
Tameness in expansions of the real field (2016, background): “Now let F ⊆ [0, 1 ]2 be closed and σ be a permutation of [0,1].”
Notes on Thin Matrix Groups (2012, methodology): “Thanks to the referee for pointing me to the relevant applications in [Gromov and Guth 2012].”
Growth in groups: ideas and perspectives (2013, unknown): “t is essentially topological [ST83]; the best known bound for the sum-product theorem over R has a direct proof, also topological [Sol09].) Theorem 3.6 (Sum-product theorem [BKT04], [BGK06]; see also [EM03]). For any A⊂ F∗ p with C<|A| <p1−ǫ, ǫ>0, we have max(|A· A|,|A+A|) >|A|1+δ, where”
On the Erdős-Volkmann and Katz-Tao ring conjectures (2003, background): “Acknowledgment. The author is indebted to W. Schlag and T. Tao for several comments on earlier versions of the manuscript and also bringing [ EM ] to my attention.”