A polynomial bound in Freiman's theorem

By Mei-Chu Chang. 2002

236 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c2a813490636c3f0232a52509679315b6cfda39a

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Numerical Analysis

Citations in the corpus

Freiman's theorem in an arbitrary abelian group (2005, background): “A complete account of the arguments of [4] and [13], together with all the necessary background, may be found in lecture notes of the first author [7].”
Growth in groups: ideas and perspectives (2013, unknown): “[GT08], among many others, and another one based on the study of growth in abelian groups, starting with Freiman [Fre73], Erd˝os-Szemer´edi [ES83] and Ruzsa [Ruz94], [Ruz99] and continuing in [GR05], [Cha02], [San12]. There has also been a vein of a more geometrical ﬂavor (e.g., [ST83]). The use of ”
The structure of approximate groups (2011, methodology): “6 is an immediate consequence of the case k = 2 of this lemma and (the continuous variant of) “Chang’s lemma” [9], which is the following statement.”
On the singularity probability of random Bernoulli matrices (2005, methodology): “A|/|P| ≥ δ. Freiman’s proof was rather complex. A cleaner proof, based on Freiman’s ideas, is presented by Bilu in [1]. Ruzsa [15] gave a diﬀerent and quite short proof; this was then reﬁned by Chang [4]. There are explicit bounds known on the values of r and δ in terms of C, but for our application”
Sets with Small Sumset and Rectification (2004, unknown): “Mei-Chu Chang, ‘A polynomial bound in Freiman’s theorem’, Duke Math.”
Open problems in additive combinatorics (2007, unknown): “We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests.”
Decompositions, approximate structure, transference, and the Hahn–Banach theorem (2008, background): “These bounds have subsequently been improved by Chang [C], who added further interesting ingredients to Ruzsa’s argument.”
Finite field models in additive combinatorics (2004, methodology): “Using a Fourier-analytic technique of Bogolyubov [8] one may show that f(α) ≪ α−2, and a refinement of this technique due to Chang [12] allows one to improve this to f(α) ≪ α−1 log(1/α).”
On Roth's theorem on progressions (2010, background): “Lemma 3.5 (The Chang bound, [San10, Lemma 4.6]).”
On the Erdős-Volkmann and Katz-Tao ring conjectures (2003, methodology): “We use the quantitative version as obtained in [ C ].”
On a question of Erdős and Moser (2005, methodology): “The same applies for Chang’s bound on Freiman’s theorem.”
Near optimal bounds in Freiman's theorem (2011, methodology): “We also remark that, using Freiman-Bilu [1] argument (see also [3]), one cannot deduce from Theorem 1 that A is contained in a progression with linear dimension and size at most exp(K1+C(logK) −1/2 )|A|.”
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. (2020, unknown): “Although we do not require Chang’s lemma itself explicitly in this paper, the underlying ideas are related to what follows.”
Long arithmetic progressions in sumsets: Thresholds and bounds (2005, background): “The most recent estimate on k (as a function of c) is due to Chang [5].”