We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Schrödinger, and the gravity and gravity-capillary water wave equations. We also discuss applications to certain nonlinear dispersive equations.

Let A ⊂ R be finite. We quantitatively improve the Balog-Wooley decomposition, that is A can be partitioned into sets B and C such thatWe use similar decompositions to improve upon various sum-product estimates. For instance, we show |A + A| + |AA| |A| 4/3+5/5277 . The author is partially supported by NSF grant DMS-1501982 and would like to thank KevinFord for financial support. The author also thanks Kevin Ford and Oliver Roche-Newton for useful comments and suggestions, as well as the referee for a meticulous and timely reading and helpful suggestions.

Let F be a field and a finite A ⊂ F be sufficiently small in terms of the characteristic p of F if p > 0.We strengthen the "threshold" sum-product inequalitydue to Roche-Newton, Rudnev and Shkredov, toas well as |AA| 36 |A − A| 24 ≫ |A| 73−o(1) .The latter inequality is "threshold-breaking", for it shows for ǫ > 0, one has |AA| |A| 1+ǫ ⇒ |A − A| ≫ |A| 3 2 +c(ǫ) , with c(ǫ) > 0 if ǫ is sufficiently small. This implies that regardless of ǫ, |AA − AA| ≫ |A| 3 2 + 1 56 −o(1) .

We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ Z one has |p · A + q · A| ≥ (p + q)|A| − (pq) (p+q−3)(p+q)+1 .

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