Georgia Institute of Technology College of Sciences

No seminar due to the "Special tea" fro the Admitted Student Day, which is 3-4 pm in the Skiles atrium.

 It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.

We shall discuss the recent breakthrough of  Annika Heckel on the chromatic number of the binomial random graph G(n,1/2),  showing that it is not concentrated on any sequence of intervals of length n^{1/4-o(1)}.

To put this into context, in 1992 Erdos (and also Bollobás in 2004) asked for any non-trivial results asserting a lack of concentration, pointing out that even the weakest such results would be of interest.  
Until recently this seemed completely out of reach, in part because there seemed to be no obvious approach/strategy how to get one's foot in the door. 
Annika Heckel has now found such an approach, based on a clever coupling idea that compares the chromatic number of G(n,1/2) for different n. 
In this informal talk we shall try to say a few words about her insightful proof approach from https://arxiv.org/abs/1906.11808

Please note the unusual room (Skiles 202)

We report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums. It turns out that sums in the
class we consider are much smaller than would be predicted by certain probabilistic heuristics. After stating the motivation, and our theorem,
we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem. For example, we prove a "Pentagonal Number Theorem for the Primes", which counts the number of primes (with von Mangoldt weight) in a set of intervals very precisely. In fact the result is  stronger than one would get using a strong form of the Prime Number Theorem and also the Riemann Hypothesis (where one naively estimates the \Psi function on each of the intervals; however, a less naive argument can give an improvement), since the widths of the intervals are smaller than \sqrt{x}, making the Riemann Hypothesis estimate "trivial".

Based on joint work with Ernie Croot.