Matrix scaling, explicit Sinkhorn limits, and arithmetic

The process of alternately row scaling and column scaling a positive $n \times n$matrix$A$convergestoadoublystochasticpositive$n \times n$matrix$S(A)$,calledthe\mathit{\text{Sinkhorn}}\mathit{\text{limit}}of$A$. Exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices are computed, and related problems in diophantine approximation are considered.