2019/02/26 by Wenbo Sun, Sun, Wenbo
Mathematics · #05D10 (Primary) 11B30 #11N37 #11N60 #11N80 #11R04 #37A45 (Secondary) #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1902.09712
openalex publication_date 2019/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We formulate the generalized Sarnak's Möbius disjointness conjecture for an arbitrary number field K, and prove a quantitative disjointness result between polynomial nilsequences (Φ(g(n)Γ))n∈ℤD and aperiodic multiplicative functions on OK, the ring of integers of K. Here D=[K\colonℚ], X=G/Γ is a nilmanifold, g\colonℤD→ G is a polynomial sequence, and Φ\colon X→ ℂ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of Kátai in OK. We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on K, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in OK to be zero; (3) we provide partition regularity results over K for a large class of homogeneous equations in three variables. For example, for a,b∈ℤ\backslash\0\, we show that for every partition of OK into finitely many cells, where K=ℚ(√(a),√(b),√(a+b)), there exist distinct and non-zero x,y belonging to the same cell and z\inOK such that ax2+by2=z2.