Novel use of ergodic theory leads to breakthrough solution of conjectures of Erdős about sumsets
IAS - April 2023
Solving thorny mathematics conjectures often takes more than hard work—it can also depend on creativity and collaboration. Over the past few years, two recent members of the Institute for Advanced Study in Princeton have employed both in their persistent effort to uncover solutions to long-standing conjectures of Erdős about sumsets.
Joel Moreira, PhD, and Florian K. Richter, PhD, focused first on solving the following question, posed by Erdős many decades ago, about sumsets: If A is a dense subset of the positive integers, are there infinite sets B and C of integers such that the sumset B + C is contained in A?
The question at face value appears rather simple to many mathematicians, but it had proven resistant to many efforts to solve it. It took a fresh approach from Moreira and Richter, along with Donald Robertson, PhD, to definitively answer the question in the affirmative.
A proof of a sumset conjecture of Erdős
Even though the conjecture is a problem in additive combinatorics, the three collaborators applied not only combinatorial and nonstandard analysis but also ultrafilters and other tools from ergodic theory in their solution. Using ergodic theory, a branch of dynamical systems that was developed to study celestial mechanics, was unusual, but it made sense to Moreira, Richter, and Robertson because all three had studied under the same dissertation advisor: Vitaly Bergelson, PhD, an ergodic theory thought leader at Ohio State University. Bergelson’s own dissertation advisor, Hillel Furstenberg, PhD, was the first to apply the machinery of ergodic theory to additive combinatorics problems.
Bergelson noted that the work by his former students represents a real breakthrough not only because they found a way to solve the conjecture but also because “they used clever new methods from functional analysis and combinatorics and with dynamical flavor, and they did it in big generality.” The solution was posted March 2018 in arxiv and published March 2019 in Annals of Mathematics (doi:10.4007/annals.2019.189.2.4)
While the work was widely admired, it actually answered the easier of two Erdős conjectures that the three wanted to solve, according to Richter. “The more difficult one we initially didn't know how to do,” he said, referring to Erdős’s conjecture that any set of positive whole numbers with positive density contains a configuration of the form B + B + t, for some t and some infinite set B.
“Once we knew how to solve the easier conjecture we wanted to solve the more difficult one, and we tried very intensively for quite a while, but we hit a wall,” Richter said.
Infinite sumsets in sets with positive density
It wasn’t until a few years later, after he and Moreira spent time as postdocs at Northwestern University under the mentorship of Bryna Kra, PhD, that they decided to pick it up again, this time with Kra as a collaborator. Building on new insights from Kra, who studies dynamical systems using symbolic dynamics, and Bernard Host, PhD, a mathematician at Université Paris-Est Marne-la-Vallée, they developed a new method in 2022 that helped generalize their original solution. They were able to prove that any set A of positive integers with positive upper Banach density contains, for any number k, a sumset of the form B1+B2+...+Bk where B1,...,Bk are all infinite sets of integers.
Soon after, they realized that they could use this new approach to solve the second, harder, Erdős conjecture.
A proof of Erdős’s B + B + t conjecture
The main novelty of this more recent work, posted to arxiv in June 2022, is that it involves infinite configurations, Moreira said. While similar results have involved finite configurations, he said, “our work was essentially the first result for infinite configurations. It opens the door to a whole new collection of possible results involving infinite configurations in general that were locked before.”
To outsiders, the two conjectures or Erdős may seem rather playful, the mathematicians noted, because they involve taking a bunch of numbers and then going looking for all the sums they can build with these numbers. But mathematicians are interested in these conjectures because the only way to solve them is to better understand the arithmetic structure of the integers.
“The deeper understanding you have of how these different arithmetic structures of the integers intertwine, the more likely you are to solve other questions related to these,” Richter said. Furthermore, it turns out that “there are very sophisticated analytic objects that you would never expect that control these kinds of sums,” he said.
Those objects, called nilpotent groups, are important in physics and geometry, and now it turns out they are also important in understanding arithmetics. “This connection to these objects and to dynamics on these objects is something very surprising, something very deep, and something that will be helpful in the future as well,” Richter said.
Everyone who worked on the conjectures is currently working in a different location. Moreira is at the Institute for Advanced Study, spending the term away from University of Warwick. Richter has finished his term at the Institute and is now at the École Polytechnique Fédérale de Lausanne. Kra is in Chicago at Northwestern University. Robertson is at University of Manchester.
Nevertheless, they are still able to collaborate. They continue to work on polishing and improving their solutions to the Erdős conjectures. They are trying to figure out how to extend their B + B + t solution to B + B + B + t. “The proof we have doesn't work in this case, so we need to come up with something new,” Moreira said.
And, because their result was one of the first instances of studying infinite—rather than finite—arithmetic structures, there are now many natural follow up questions that they can start to answer. As Richter said, “Based on the advanced understanding of the additive structure of the integers that our work provided, there are related questions that can now be pursued.”