40 research outputs found
Automatic sets of rational numbers
The notion of a k-automatic set of integers is well-studied. We develop a new
notion - the k-automatic set of rational numbers - and prove basic properties
of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
Counting Algebraic Units with Bounded Height
AbstractLet K be an algebraic number field of degree d and let U denote its group of units. Suppose U has rank r and regulator R. Let U(q) denote the number of units in U with height less than q. We obtain an asymptotic formula for U(q) of the shape U(q) = A(log q)r + O((log q)r − 1 − δ), where δ is given explicitly in terms of d and R