Higher-dimensional projection constructions

Establish Diophantine and lattice-geometric bounds for higher-dimensional coordinate projections by replacing the rational projective line \(P^1(\mathbb{Q})\) with an appropriate higher-dimensional projective space and imposing non-degeneracy conditions on larger subsets of coefficient vectors; determine effective constructions with favorable mesh-ratio constants.

Background

The paper’s two algebraic constructions are designed specifically to guarantee quasi-uniformity of all two-dimensional coordinate projections. Their fixed algebraic degrees also imply that higher-dimensional projections generally need not be uniformly distributed.

The conclusion identifies a concrete extension: use number fields of higher degree and higher-dimensional rational coefficient configurations to address higher-dimensional projections. The required Diophantine estimates, lattice-geometric bounds, and effective constructions with favorable mesh-ratio constants are not established in the paper and are explicitly deferred to future work.

References

The present paper has focused exclusively on two-dimensional coordinate projections. The underlying algebraic mechanism suggests a possible extension to higher-dimensional projections by using number fields of higher degree together with higher-dimensional rational coefficient configurations. For instance, one would naturally replace P1(Q) by an appropriate higher-dimensional projective space and impose non-degeneracy conditions on larger subsets of coefficient vectors. Establishing the corresponding Diophantine and lattice-geometric bounds, and determining effective constructions with favorable mesh-ratio constants, are beyond the scope of the present paper and are left for future work.

— Algebraic constructions of point sequences with quasi-uniform two-dimensional projections  (2608.12690 - Goda, 13 Aug 2026) in Section 7, Concluding remarks