arXiv subjects

Sair Shaikh

Publications and source records attributed to Sair Shaikh.

3 recordsLinked to original sources

arXiv2026

Pointless K3 surfaces over finite fields

We prove that K3 surfaces with no rational points over the finite field with $q$ elements exist if and only if $q \leq 11$. Our approach applies linear programming techniques to improve on generic sphere packing bounds for the discriminants of even rootless definite lattices. These bounds are applied to constrain the possible zeta functions of K3 surfaces with no rational points via the Artin--Tate formula. The remaining cases are ruled out by direct analysis of primitive embeddings of particular lattices into Niemeier lattices.

math.NT↗

arXiv2026

A Sharper Picture of Generalization in Transformers

We study transformers' generalization behavior on boolean domains from the perspective of the Fourier spectra of their target functions. In contrast to prior work (Edelman et al., 2022; Trauger & Tosh, 2024), which derived generalization bounds from Rademacher complexity, we investigate the feasibility of obtaining generalization bounds via PAC-Bayes theory. We show that sparse spectra concentrated on low-degree components enable low-sharpness constructions with good generalization properties. Our idea is to show the existence of flat minima implementing any boolean function of sparsity no greater than the context length, and then apply a PAC-Bayes bound to an idealized low-sharpness learner, resulting in a non-vacuous generalization bound. We use this to give a formal account of why chain-of-thought improves generalization for high-degree target functions, and show that the complexity parameters in our bound can be efficiently estimated via property testing. We evaluate predictions empirically and conduct a mechanistic interpretability study to support the realism of our theoretical construction in real transformers.

cs.LG↗

arXiv2025

Strata of toric hyperplane arrangements, zonotope lattice points, and the Bondal-Thomsen collection

We show that strata of oriented toric hyperplane arrangements are in bijection with a collection of lattice points in a zonotope. Moreover, we relate the dimension of the stratum and the dimension of the minimal face of the zonotope containing the corresponding lattice point. We discuss how this correspondence is related to toric varieties and the Bondal-Thomsen generators of their derived categories.

math.CO↗