The Favard length of a planar set E is the average length of its one-dimensional projections. In a joint project with Bond and Volberg, we prove new upper bounds on the decay of the Favard length of finite iterations of 1-dimensional planar Cantor sets with a rational product structure. This improves on the earlier work of Nazarov-Peres-Volberg, Bond-Volberg, and Laba-Zhai, and introduces new algebraic and number-theoretic methods to this area of research. The estimates are of interest in geometric measure theory, ergodic theory and analytic function theory.