A new view on HJLS and PSLQ: sums and projections of lattices

Abstract

The HJLS and PSLQ algorithms are the de facto standards for discovering non-trivial integer relations between a given tuple of real numbers. In this work, we provide a new interpretation of these algorithms, in a more general and powerful algebraic setup: we view them as special cases of algorithms that compute the intersection between a lattice and a vector subspace. Further, we extract from them the first algorithm for manipulating finitely generated additive subgroups of a euclidean space, including projections of lattices and finite sums of lattices. We adapt the analyses of HJLS and PSLQ to derive correctness and convergence guarantees.