Generalizations of Pollard’s Theorem to General Abelian Groups
Location
Room 101, Hume Hall
Start Date
27-4-2024 1:00 PM
End Date
27-4-2024 2:00 PM
Description
Let A+t B denote the set of elements in A+B with at least t representations in the form a+b, where a ∈ A and b ∈ B. Pollard’s theorem shows the bound t i=1|A+iB| ≥ t·min{p,|A|+|B|−t} in Cp with p prime. For general abelian groups, Grynkiewicz gave a Kneser-type theorem, which provides strong structural information when t i=1 |A +i B| < t|A| + t|B| −2t2 +3t−2. Another theorem is given by Hamidoune and Serra, in which the structural result is weaker. In this talk, we will also show a recent result, which optimizes the coefficient of the quadratic term in Grynkiewicz’s theorem, and still preserves all its structural results. Joint work with David J. Grynkiewicz. [A formatted abstract is also attached.]
Relational Format
conference proceeding
Recommended Citation
Wang, Runze, "Generalizations of Pollard’s Theorem to General Abelian Groups" (2024). AMS-AWM Student Research Symposium. 19.
https://egrove.olemiss.edu/ams_awm_sympo/2024/schedule/19
Generalizations of Pollard’s Theorem to General Abelian Groups
Room 101, Hume Hall
Let A+t B denote the set of elements in A+B with at least t representations in the form a+b, where a ∈ A and b ∈ B. Pollard’s theorem shows the bound t i=1|A+iB| ≥ t·min{p,|A|+|B|−t} in Cp with p prime. For general abelian groups, Grynkiewicz gave a Kneser-type theorem, which provides strong structural information when t i=1 |A +i B| < t|A| + t|B| −2t2 +3t−2. Another theorem is given by Hamidoune and Serra, in which the structural result is weaker. In this talk, we will also show a recent result, which optimizes the coefficient of the quadratic term in Grynkiewicz’s theorem, and still preserves all its structural results. Joint work with David J. Grynkiewicz. [A formatted abstract is also attached.]
Comments
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