### Diophantine approximation and deformation, by M. Kim, D. Thakur, J. F. Voloch

We associate certain curves over function fields to given algebraic power
series and show that bounds on the rank of Kodaira-Spencer map of this curves
imply bounds on the exponents of the power series, with more generic curves
giving lower exponents. If we transport Vojta's conjecture on height inequality
to finite characteristic by modifying it by adding suitable deformation
theoretic condition, then we see that the numbers giving rise to general
curves approach Roth's bound. We also prove a hierarchy of exponent bounds
for approximation by algebraic quantities of bounded degree.

M. Kim, D. Thakur, J. F. Voloch <kim@math.arizona.edu, thakur@math.arizona.edu, voloch@math.utexas.edu>