We give a criterion implying subcritical behavior for quasi-periodic Schrödinger operators where the potential sampling function is given by a trigonometric polynomial. Subcritical behavior, in the sense of Avila's global theory, is known to imply purely absolutely continuous spectrum for all irrational frequencies and all phases.

ABSTRACT. Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A − A| > |A + A| for a finite set A. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of {0, . . . , n} are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as n → ∞. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in A to go to zero, have been analyzed extensively.We consider many of these problems in a more general setting. Instead of just one set A, we study sums and differences of pairs of correlated sets (A, B). Specifically, we place each element a ∈ {0, . . . , n} in A with probability p, while a goes in B with probability ρ 1 if a ∈ A and probability ρ 2 if a ∈ A. If |A + B| > |(A − B) ∪ (B − A)|, we call the pair (A, B) a sum-dominant (p, ρ 1 , ρ 2 )-pair. We prove that for any fixed ρ = (p, ρ 1 , ρ 2 ) in (0, 1) 3 , (A, B) is a sum-dominant (p, ρ 1 , ρ 2 )-pair with positive probability, and show that this probability approaches a limit P ( ρ). Furthermore, we show that the limit function P ( ρ) is continuous. We also investigate what happens as p decays with n, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.

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A sum-dominant set is a finite set A of integers such that |A + A| > |A − A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N 0 , the proportion ρ s,d n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {−n, . . . , n} also remains positive in the limit.We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) − L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit. 19 6. Future Directions 21 Appendix A. Number of Pairs of Strictly Antipodal Vertices 22 References

We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines all previously known bounds of this type. We also improve Nisan and Szegedy's well-known inequality bs(f ) ≤ deg(f ) 2 by a constant factor, thereby improving Huang's recent proof of the sensitivity conjecture by the same constant.