Paley ETF satisfies RIP beyond the square-root bottleneck

Prove that the Paley equiangular tight frame in C^{(p+1)/2} with n = p + 1 vectors satisfies the Restricted Isometry Property at sparsity m ≍ p^{1/2+ε} for some ε > 0, i.e., any (p^{1/2+ε})-column submatrix has uniformly bounded condition number independent of p.

Background

The Paley ETF arises from quadratic residues in the DFT and forms a deterministic equiangular tight frame. RIP beyond the √n bottleneck is a long-standing challenge for deterministic constructions; Paley ETFs are a prime candidate under pseudorandomness heuristics.

A proof would be a breakthrough in deterministic compressed sensing, tightly connecting pseudorandom graphs and sparse recovery.

References

Conjecture The Paley Equiangular Tight Frame construction satisfies the Restricted Isometry Property beyond the square-root bottleneck. In other words, there exists $\varepsilon>0$ such that, for $m \sim p{1/2+\varepsilon}$, any matrix formed by picking $m$ (distinct) vectors in the Paley ETF has condition number bounded by a constant independent of $p$.

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “On the clique number of the Paley Graph (ASB)” (Entry 12)