Sarnak's Conjecture for nilsequences on arbitrary number fields and applications
| dc.contributor.author | Sun, Wenbo | en |
| dc.date.accessioned | 2024-01-24T14:26:36Z | en |
| dc.date.available | 2024-01-24T14:26:36Z | en |
| dc.date.issued | 2023-02-15 | en |
| dc.description.abstract | 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:Q], X=G/Γ is a nilmanifold, g:ZD→G is a polynomial sequence, and Φ:X→C is a Lipschitz function. This result, being a generalization of a previous theorem of the author in [44], requires a significantly different approach, which involves with 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﹨{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∈OK such that ax2+by2=z2. | en |
| dc.description.version | Accepted version | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier | 108883 (Article number) | en |
| dc.identifier.doi | https://doi.org/10.1016/j.aim.2023.108883 | en |
| dc.identifier.eissn | 1090-2082 | en |
| dc.identifier.issn | 0001-8708 | en |
| dc.identifier.orcid | Sun, Wenbo [0000-0003-3399-3937] | en |
| dc.identifier.uri | https://hdl.handle.net/10919/117642 | en |
| dc.identifier.volume | 415 | en |
| dc.language.iso | en | en |
| dc.publisher | Elsevier | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.title | Sarnak's Conjecture for nilsequences on arbitrary number fields and applications | en |
| dc.title.serial | Advances in Mathematics | en |
| dc.type | Article - Refereed | en |
| dc.type.dcmitype | Text | en |
| dc.type.other | Journal Article | en |
| pubs.organisational-group | /Virginia Tech | en |
| pubs.organisational-group | /Virginia Tech/Science | en |
| pubs.organisational-group | /Virginia Tech/Science/Mathematics | en |
| pubs.organisational-group | /Virginia Tech/All T&R Faculty | en |
| pubs.organisational-group | /Virginia Tech/Science/COS T&R Faculty | en |