The study of Diophantine equations encompasses a diverse portion of modern number theory. Recent years have seen spectacular progress on solving linear Diophantine equations in certain sets of interest, such as dense sets or the set of primes. Much of this progress has been achieved by breaking the problem down into a structure versus randomness dichotomy, using tools from additive combinatorics. One tackles the structured problem using techniques from classical analytic number theory and dynamical systems, whilst the ‘random' problem is handled using ideas informed by probabilistic combinatorics and Fourier analysis.
The consequences of this rapidly developing theory for non-linear Diophantine equations have yet to be fully explored. Some possible research topics include (but are not limited to) the following:
- Existence of solutions to systems of Diophantine equations in dense sets. To what extent can Szemerédi’s theorem be generalised to non-linear systems of equations?
- Quantitative bounds for sets lacking Diophantine configurations. Can one obtain good quantitative bounds in the polynomial Szemerédi theorem? What about sets lacking progressions with common difference equal to a prime minus one?
- Partition regularity of Diophantine equations. Can one generalise a Ramsey-theoretic criterion of Rado to systems of degree greater than one?
- Higher order Fourier analysis of non-linear equations. Is it possible to count solutions to hitherto intractable Diophantine equations by developing the Hardy—Littlewood method along the lines of Green and Tao? What are the obstructions to uniformity for such equations?