Kleinbock, D., Moshchevitin, N., Warren, J. M., & Weiss, B. (2026). Singularity, weighted uniform approximation, intersections and rates. Compositio Mathematica, 161(11), 2990–3016. https://doi.org/10.1112/S0010437X25102777
E104-05 - Forschungsbereich Kombinatorik und Algorithmen
-
Journal:
Compositio Mathematica
-
ISSN:
0010-437X
-
Date (published):
8-Jan-2026
-
Number of Pages:
27
-
Publisher:
CAMBRIDGE UNIV PRESS
-
Peer reviewed:
Yes
-
Keywords:
Diophantine Approximation; singular matrices; approximation on manifolds; transference principle
en
Abstract:
A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was
subsequently revisited and extended by many authors. In the present paper we adapt Khintchine's argument to show that the sets of very singular matrices and their weighted analogues intersect many manifolds and fractals, and
have strong intersection properties. We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in n-dimensional space.
en
Project title:
Geometrie der Zahlen in Diophantischer Approximation: PAT1961524 (FWF - Österr. Wissenschaftsfonds)
