A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A ⊂ Z such that |A + A| < |A − A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant [MO] proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that |ǫ 1 A + · · · + ǫ k A| > |δ 1 A + · · · + δ k A| a positive percent of the time for all nontrivial choices of ǫ j , δ j ∈ {−1, 1}. Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets.We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken. For example, we prove that for any m, |ǫ 1 A+· · ·+ǫ k A|−|δ 1 A+· · ·+δ k A| = m a positive percentage of the time. We find the limiting behavior of kA = A + · · · + A for an arbitrary set A as k → ∞ and an upper bound of k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if A, A + A, . . . , kA are all sum-dominant. Numerical searches were unable to find even a 2-generational set (heuristics indicate that the probability is at most 10 −9 , and quite likely significantly less). We prove that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k.
The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions φ . We prove similar results for families of cuspidal Maass forms, the other natural family of GL 2 /Q L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp( φ ) ⊆ (−3/2, 3/2) when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1, 1), though we still uniquely specify the symmetry type by computing the 2-level density.
We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and 'explicit' constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are k-generational sum-dominant (meaning A, A + A, . . . , kA = A + · · · + A are each sum-dominant).
We study low-lying zeroes of L-functions and their n-level density, which relies on a smooth test function φ whose Fourier transform φ has compact support. Assuming the generalized Riemann hypothesis, we compute the n th centered moments of the 1-level density of low-lying zeroes of L-functions associated with weight k, prime level N cuspidal newforms as N → ∞, where supp( φ) ⊂ (−2/n, 2/n). The Katz-Sarnak density conjecture predicts that the n-level density of certain families of L-functions is the same as the distribution of eigenvalues of corresponding families of orthogonal random matrices. We prove that the Katz-Sarnak density conjecture holds for the n th centered moment of the 1-level density for test functions with φ supported in (−2/n, 2/n), for families of cuspidal newforms split by the sign of their functional equations. Our work provides better bounds on the percent of forms vanishing to a certain order at the central point. Previous work handled the 1-level for support up to 2 and the n-level up to min(2/n, 1/(n − 1)); we are able to remove the second restriction on the support and extend the result to what one would expect, based on the 1-level, by finding a tractable vantage to evaluate the combinatorial zoo of terms which emerge.
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