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In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p -adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p -adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets. [ABSTRACT FROM AUTHOR]

Text 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 sum-dominant 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 n s , d of subsets of { 0 , … , n } that are missing exactly s sums in { 0 , … , 2 n } and exactly 2 d 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 n s , d now denote the proportion of subsets of L ( n P ) that are missing exactly s sums in L ( n P ) + L ( n P ) and exactly 2 d differences in L ( n P ) − L ( n P ) . As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ n s , d . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ n s , d 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 ( n P ) that are missing exactly s sums and at least 2 d 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 ( n P ) also remains positive in the limit. Video For a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc . [ABSTRACT FROM AUTHOR]

Abstract: Digital Kronecker-sequences are a non-archimedean analog of classical Kronecker-sequences whose construction is based on Laurent series over a finite field. In this paper it is shown that for almost all digital Kronecker-sequences the star discrepancy satisfies for infinitely many , where only depends on the dimension s and on the order q of the underlying finite field, but not on N. This result shows that a corresponding metrical upper bound due to Larcher is up to some term best possible. [Copyright &y& Elsevier]

Abstract: Text: A More Sums Than Differences (or sum-dominant) set is a finite set with . Though it was believed that the percentage of subsets of that are sum-dominant tends to zero, Martin and OʼBryant proved a positive percentage is sum-dominant. We generalize their result to other sums and differences. We prove that a positive percent of the time for all nontrivial choices of , and give explicit constructions. We construct sets exhibiting different behavior as more sums/differences are taken. We prove that for any m, a positive percentage of the time. We find the limiting behavior of for an arbitrary set A as and an upper bound on k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if are all sum-dominant. We prove that for any k a positive percentage of sets is k-generational, and no set is k-generational for all k. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ERSMlrEAijY;list=UUfJicAn0WSCOS0IZWMy7HsA;index=1;feature=plcp. [Copyright &y& Elsevier]