Near optimal three-fold additive energy bound for points on convex curves

What the math gives to ML

The paper proves that lifting a finite scalar set onto any strictly convex curve γ(t)=(t,f(t)) produces near-minimal three-fold additive energy: the number of six-tuples with equal sums of three lifted points is only |X|^{3+ε}, up to the unavoidable diagonal contribution. This suggests structured anti-collision codebooks for neural representations, especially when downstream operations add or average embeddings and representational aliasing is harmful. The most direct experiment is to replace learned scalar or discrete-code embeddings with a fixed convex lift such as (t,t²), then measure exact and approximate collisions and compositional retrieval accuracy against unconstrained and random embeddings.