# On Szemer\'edi's theorem with differences from a random set

## TL;DR

This paper investigates Szemerédi's theorem with restricted, random differences over integers and finite fields, linking thresholds to conjectures about dual functions and nilsequences, revealing different behaviors in these settings.

## Contribution

It connects the threshold for random differences in Szemerédi's theorem to dual function approximations by nilsequences and compares integer and finite field cases.

## Key findings

- Threshold over integers depends on dual function approximation conjecture.
- Threshold over finite fields differs from that over integers.
- Provides new insights into the behavior of arithmetic progressions with random differences.

## Abstract

We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for $\mathbb{P}(d \in S)$ for Szemer\'{e}di's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.