Recall from earlier posts Gil’s modular conjecture for HAPs. It states that if is large enough and is a function from to that never takes the value 0, then for every there exists a HAP such that mod . It is easy to see that this implies EDP, so it may well be very hard, or even false. However, one can hold out a little hope that, as with some strengthenings of statements, it is in fact easier, because it is in some way more symmetrical and fundamental. Given that, it makes good sense, as Gil has suggested, to try to prove modular versions of known discrepancy theorems, in the hope of developing general techniques that can then be tried out on the modular EDP conjecture.
A very obvious candidate for a discrepancy theorem that we could try to modularize is Roth’s theorem, which asserts that for any -valued function on there exists an arithmetic progression such that . That gives rise to the following problem.
Problem. Let be a prime. What is the smallest such that for every function that never takes the value 0, every can be expressed as for some arithmetic progression ?
In this post I shall collect together a few simple observations about this question.