# Seminar

## Rocky Mountain Algebraic Combinatorics Seminar

### Structure in sumsets I

John Griesmer

Given two subsets A, B of an abelian group, their sumset is A + B: = {a+b : ∈ A, bB}. Their study is divided roughly into direct theorems, which deduce properties of A+B from hypotheses on A and B, and inverse theorems, which deduce properties of A and B from hypotheses on A and B.

A prototypical direct theorem is the Cauchy-Davenport theorem, which states that if p is prime and A, B are nonempty subsets of \mathbbZ/p\mathbb Z, then |A+B| ≥ min{|A|+|B|−1, p}. The corresponding inverse theorem, due to A. G. Vosper, provides a simple classification of the pairs A, B where equality occurs.
Another classical result is the Steinhaus lemma, which says that if A,B ⊂ \mathbb R both have positive Lebesgue measure, then A+B contains an interval.
We will survey these and other classical results, together with their proof techniques, including the basic theory of Fourier analysis as it applies to sumsets. We will also discuss connections to other results in additive combinatorics, such as the recent Kelley-Meka breakthrough on counting three-term arithmetic progressions in sets of integers. No background in measure theory is required.

### Structure in sumsets II

John Griesmer

Given two subsets A, B of an abelian group, their sumset is A + B: = {a+b : ∈ A, b ∈ B}. An inverse theorem on sumsets derives structural results on A and B from hypotheses on A+B. For example, if A and B are nonempty sets of integers with |A+B| < |A|+|B|, then A and B are arithmetic progressions with the same common difference, or |A|=1 or |B|=1. We will survey some of the many recent results determining the structure of pairs A and B where |A+B| is "slightly larger" than |A|+|B| (in various senses).

This will motivate the following general problem: how can one construct sets A and B where A+B lacks some prescribed structure? For example, given K, N ∈ \mathbb N, how can one construct a set A ⊂ (\mathbbZ/2\mathbb Z)N with |A| ≈ \frac122N, where A+A contains no subgroup of index at most K? Igor Kriz and Imre Ruzsa independently developed the same construction of such an example, variations of which are the only known way large subsets of finite abelian groups whose sumset lacks some prescribed structure. One aim of this talk is to advertise the problem of either finding a fundamentally different example, or proving that all such examples are essentially the same as Kriz and Ruzsa's.

Weber 223
4–6 pm
Friday, March 3, 2023
(Refreshments 3:30–4 pm)

This is a joint Denver U / UC Boulder / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.

### Previous Seminars:

February 17, 2023
Douglas McKenna, Dheer Desai
February 3, 2022
Richard Green, Richard Green
Nov 18, 2022
Martha Precup,
Nov 4, 2022
Martin Kassabov, Martin Kassabov
Oct 28, 2022
Aram Bingham, Aram Bingham
October 3, 2022
Nathan Lindzey, Nathan Lindzey
Sept 30, 2022
Mandi Schaeffer Fry, Jay Taylor
Sept 16, 2022
May 6, 2022
Emily King, Lucas Gagnon
Apr 8, 2022
Michael Levet,
Apr 7, 2022
Willen De Graaf,
Dec 2, 2021
Leonard Soicher,
Dec 3, 2021
Foster Tom,
Oct 28, 2021
Simon Blackburn,
Oct 15, 2021
Khanh Nguyen,
Oct 15, 2021
Khanh Nguyen,
Oct 1, 2021
Mikl\'os B\'ona, Mei Yin
Sep 17, 2021
Nathan Lindzey, James Wilson
Sep 3, 2021
Sean Griffin, Maria Gillespie
May 7, 2021
April 23, 2021
Jonah Blasiak,
April 9, 2021
Kenneth McLaughlin,
March 12, 2021
Marni Mishna,
February 26, 2020
Justin Lynd,
February 12, 2020
Michael Epstein,
December 4, 2020
Fatma Karaoglu,
November 13, 2020
James Wilson, James Wilson
October 30, 2020
Shawn Burkett, Wencin Poh
October 16, 2020
Edinah K. Gnang, Alexander Hulpke
September 18, 2020
Aram Dermenjian,
September 18, 2020
Aram Dermenjian,
August 14 2020
Federico Castillo,
July 24 2020
James Wilson,
July 10 2020
Nathan Lindsey,
June 12, 2020
Maria Gillespie,
May 29, 2020
May 8, 2020
Dustin Mixon,
Apr 24, 2020
Peter Cameron,
Apr 3, 2020
Alexander Hulpke,
Feb 28, 2019
Joseph Iverson, John Jasper
Jan 24, 2019
Jonathan Novak,
Dec 6, 2019
Nov 15, 2019
Joshua Grochow, Maria Gillespie
Nov 1, 2019
Eilidh McKemmie, Michael Epstein
Oct 11, 2019
Martin Kassabov, Eric Moorhouse
Sep 27, 2019
Alexander Hulpke, Tianjuan Xu
Sep 13, 2019
Laura Scull, Emily King
Sep 28, 2018
Sep 7, 2018
James Wilson, Alexander Hulpke
Apr 20, 2018
Maria Monks Gillespie, Alistair Savage
Apr 6, 2018
Susan Hermiller, Nathaniel Thiem
March 23, 2018
Henry Tucker, Richard Green
March 2, 2018
Nik Ruskuc, Joshua Grochow
February 16, 2018
Hamidreza Chitsaz, Mark Lewis
February 2, 2018
James B. Wilson, Anton Betten
November 10, 2017
Michael Kinyon, Petr Vojt\v{e}chovsk\'y
October 20, 2017
Eric Moorhouse, William DeMeo
October 6, 2017
Spencer Gerhardt, Joshua Grochow
September 22, 2017
Amanda Schaeffer Fry, Henry Kvinge
September 8, 2017
Peter Mayr, James B. Wilson
May 5, 2017
Ferdinand Ihringer, Gavin King
April 21, 2017
April 7, 2017
Jason Williford, Anton Betten
March 24, 2017
Isabella Novik, Peter Brooksbank
March 3, 2017
Jintai Ding, Curtis Bennett
February 17, 2017
Fatma Karaoglu, Eric Moorhouse
February 3, 2017
Tim Penttila, James B. Wilson
December 2, 2016
Jim Fowler, Andrew Kelley
November 11, 2016
Joseph Gersch, Joshua Maglione
October 28, 2016
October 14, 2016
JM Landsberg, James B. Wilson
September 30, 2016
Alexander Hulpke, Oscar Levin
September 16, 2016
Delaram Kahrobaei, Amit Patel
June 23, 2016
April 29, 2016
Nick Loehr, Jason Williford
April 15, 2016
Alexander Hulpke, Klaus Lux
April 1, 2016
Eamonn O'Brien, Izabella Stuhl
February 19, 2015
James Wilson, Anton Betten
December 4, 2015
Maria Monks Gillespie, Dane Flannery
November 13, 2015
Richard Green, Tim Penttila
October 23, 2015
Christina Boucher, Sylvia Hobart
October 9, 2015
Josh Maglione, Ghodratollah Aalipour
September 25, 2015
September 11, 2015
James B. Wilson, Tim Penttila
May 8, 2015
Amanda Schaeffer Fry, Peter Brooksbank
April 24, 2015
March 6, 2015
Felice Manganiello, Eric Moorhouse
February 20, 2015
Anton Dzhamay, Anton Betten
February 6, 2015
Alexander Hulpke, Morgan Rodgers
December 5, 2014
Stefaan De Winter, Gretchen Matthews
November 14, 2014
Greg Coxson, Tom Dorsey
October 31, 2014
Octavio Paez Osuna, Sylvia Hobart
October 10, 2014
Takunari Miyazaki, Eric Moorhouse
September 26, 2014
Elissa Ross, Anton Betten
September 12, 2014
Petr Vojtěchovský, Alexander Hulpke
May 9, 2014
Philip DeOrsey, Tim Penttila
April 25, 2014
William J Martin, Jason Williford
April 11, 2014
Victor Pambuccian, George Shakan
March 7, 2014
Nathan Lindzey, Jens Harlander
February 21, 2014
Ross McConnell, Anton Betten
November 22, 2013
Justin Hughes, Josh Maglione

Department of Mathematics