2022/01/03 by H. A. Helfgott, Harald Andrés Helfgott, Helfgott, Harald Andrés
Engineering · Mathematics · #05C48 #05C81 #11N35 #11N37 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Number Theory (math.NT) #graph theory and CDMA systems #math.CO #math.NT #msc:05C48 #msc:05C81 #msc:11N35 #msc:11N37
paper · pdf · doi:10.48550/arxiv.2201.00799
40 pages, 11 figures
arxiv created 2022/01/03 · openalex publication_date 2022/01/03 · arxiv updated 2022/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
After seeing how questions on the finer distribution of prime factorization -- considered inaccessible until recently -- reduce to bounding the norm of an operator defined on a graph describing factorization, we will show how to bound that norm. In essence, the graph is a strong local expander, with all eigenvalues bounded by a constant factor times the theoretical minimum (i.e., the eigenvalue bound corresponding to Ramanujan graphs). The proof will take us on a walk from graph theory to linear algebra and the geometry of numbers, and back to graph theory, aided, along the way, by a generalized sieve. This is an expository paper; the full proof has appeared as a joint preprint with M. Radziwi\l\l.