Recovering pulsar periodicity from time of arrival data by finding the shortest vector in a lattice

The strict periodicity of pulsars is the primary source of information we have to learn about their nature and environment, it allows us to challenge general relativity and measure gravitational waves. Identifying such a periodicity from a discrete set of arrival times is a difficult algorithmic problem, particularly when the pulsar is in a binary system. This challenge is especially acute in $γ$-ray pulsar astronomy as there are hundreds of unassociated Fermi-LAT sources awaiting a timing solution that will reveal their nature, and may allow adding them to pulsar timing arrays. The same issue arises when attempting to recover a strict periodicity for repeating fast radio bursts (FRBs). Such a detection would be a major breakthrough, providing us with the FRB source's age, magnetic field, and binary orbit. The problem of recovering a timing solution from sparse time-of-arrival (TOA) data is currently unsolvable for pulsars in binary systems and incredibly hard even for single pulsars. In a series of papers, we will develop an algorithmic set of tools that will allow us to solve the timing recovery problem under different regimes. In this paper, we frame the timing recovery problem as the problem of finding a short vector in a lattice and obtain the solution using off-the-shelf lattice reduction and sieving techniques. As a proof of concept, we solve PSR J0318+0253, a millisecond $γ$-ray pulsar discovered by FAST in a $γ$-ray directed search, in a few CPU-minutes. We discuss the assumptions of the standard lattice techniques and quantify their performance and limitations.