2024/02/20 by Stefan Steinerberger, Steinerberger, Stefan
Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2402.13202
openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Hadamard matrix is a scaled orthogonal matrix with ± 1 entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when n is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when n > 4. Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal 0< c < C < ∞ so that for all n ≥ 1, there is a matrix A ∈ \-1,1\n × n satisfying, for all x ∈ ℝn, c √(n) ‖x‖2 ≤ ‖Ax‖2 ≤ C √(n) ‖x‖2. We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all n ≥ 1.