Freddie Illingworth

Linear Diophantine equations: density and supersaturation

The Turán problem for a linear equation E (such as a + b = 2c), introduced by Ruzsa, asks the following question. How large is the largest subset of {1,...,n} that does not contain a solution to E where all the variables are distinct? For some equations, such as the 3-term arithmetic progression equation a + b = 2c, the answer is n^{1 - o(1)} while for others, such as the Sidon equation a + b = c + d, the answer is n^{1 - \alpha} (fixed \alpha > 0). There is still no characterisation of which equations have each of these behaviours.
I will describe some progress on this problem and discuss the many things we don't know.

Based on joint work with António Girão (UC London), Eoin Hurley (LSE), and Lukas Michel (ETH Zürich).