Entropy methods in Additive Combinatorics and Analytic Number Theory

This project falls within the EPSRC Logic and Combinatorics, and EPSRC Number Theory research areas.
The solutions to many problems in additive combinatorics and analytic number theory require a separation between structure and randomness, between regular and uniform behaviors. Indeed, consider a general problem of estimating (weighted) counts of a certain pattern (such as primes, prime tuples or arithmetic progressions) in a certain space (an interval, a set with large density, a graph). In the pseudorandom case, when the pattern and the space are roughly independent, the joint count can be estimated as the product of two individual counts. The other extreme case, when the space is highly structured with respect to the pattern, is often either easy to analyze for different reasons, or rare. The strategy, therefore, is to decompose the general case into a structured and a random part, or into multiple parts which cannot all be structured simultaneously, and then to deal with these parts individually.
Information theory, based on the notion of Shannon entropy, provides a convenient language for formalizing such arguments. Shannon originally introduced entropy as a tool in coding theory, giving limits for data compression; since then, his ideas found applications across cryptography, statistical inference and bioinformatics, among others. More recently, though, the value of information theory has also been recognized in pure mathematics.
For instance, Szemerédi's theorem, one of the cornerstone results of additive combinatorics, states that every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions; it has an impressive variety of proofs, all of which rely on the aforementioned decomposition of the given set into structured and random parts. In the graph-theoretic approach, the tool used to accomplish this decomposition is Szemerédi's regularity lemma, which has an information-theoretic analogue based on an entropy increment argument due to Tao; this formulation is both slightly simpler and more general.
Similarly, Tao's solution to the famous Erdös discrepancy problem employed an innovative entropy decrement argument. Roughly speaking, since variations in Shannon entropy are expressible as (conditional) mutual information, which measures approximate (conditional) independence, entropy increment and decrement algorithms can find a scale at which (or the optimal extent to which) two random variables exhibit weak independence properties. In Tao's work, this led to a logarithmically averaged version of Elliott's conjecture, which was enough to answer Erdös's question.

The main goal of this project is to extend the applications of entropy methods to various combinatorial and arithmetic problems.
One objective is to obtain a fully information-theoretic proof of Szemerédi's theorem, using generalizations of Tao's regularity lemma. This formulation may also simplify the transference principle machinery used to prove the Green-Tao theorem (stating that the primes contain arbitrarily long arithmetic progressions), which currently uses Szemerédi's theorem as a black-box.
Since information-theoretic quantities can function as substitutes for Gowers norms, it would also be desirable to translate the Fourier-analytic approach to Szemerédi's theorem into this language. If such an approach is successful for arithmetic progressions, one would hope to extend it to the case of suitable polynomial progressions, at first by adapting Sarah Peluse's degree-lowering argument.
Another potential direction, stemming from the solution to the Erdös discrepancy problem, concerns applications of entropy methods to purely arithmetic questions such as variations of Chowla's conjecture, or related to covering systems.