Masonʹs theorem and syzygy gaps

From Masonʹs theorem on rational function fields (the progenitor of the abc-conjecture) we immediately derive upper bounds for “syzygy gaps” (Theorems 3, 8, 11, and Corollary 9). These in turn quickly give: (1) The authorʹs conjecture Z(l), used in the study of Hilbert–Kunz series. (2) A lower bound for certain “F-pure thresholds.” (3) Hanʹs explicit description of the 3-dimensional p-fractal attached to x, y and x+y. (4) Some apparently simple degree estimates, observed in the past but unproved until now (Corollary 6 and Theorem 10).