Abstract Boshernitzan The central subject in this proposal is diophantine properties of iet's (interval exchange transformations). Boshernitan's approach in the study of iet's, contrary to the 'global' approaches of others (H.Masur, W.Veech etc.), makes special emphasis on the way diophantine and algebraic properties of the parameters involved may affect the dynamical properties of an iet. Many results in analytical number theory are extendable to iet's. For example, Boshernitzan's recent result, joint with C.Carroll, that minimal iet's over a quadratic field must be pseudo-Anosof (self-similar) can be viewed as an extension of the classical Lagrange's result on the periodicity of the continued fraction representations of quadratic irrationals. Perhaps the most outstanding conjecture in this direction is obtained from an attempt to extend to iet's the famous Roth's theorem on the rate algebraic irrationals can be approximated by the rationals. (The analogue of the corresponding Liouville's theorem is true.) One of the tools at Boshernitzan's disposal in tackling the above problems is his recent discovery of a connection between the rate of recurrence of typical orbits in dynamical systems and the Hausdorff dimension of the space (quantitative version of Poincare recurrence theorem). For iet's, this connection works both directions. The same is true in many other situations (Y.Peres, D.Ornstein, B.Weiss). The list of other Boshernitzan's interests includes number theory, billiards in polygons, ergodic theorems along subsequences of integers, Hardy fields (joint projects with M.Wierdl, D.Berend, G.Kolesnik, I.Kornfeld and V.Bergelson). Because of the basic nature of the present research proposal, it is hard at this point to assess accurately the fields of possible applications. Among the subject which may benefit from our research I would only mention the possibility of creation of new code compression schemes useful in computer technology ( related to our discovery of quantitative version of Poincare recurrence theorem) and possible applications to laser technology (research with I. Kornfeld on interval translation maps).
