Problems in additive number theory, I
By Melvyn B. Nathanson. 2006
48 citations recorded by the source corpus.
Talk at the Atelier en combinatoire additive (Workshop on Arithmetic Combinatorics) at the Centre de recherches mathématiques at the Université de Montréal on April 8, 2006. Definition 1. A problem is a problem I cannot solve, not necessarily an unsolved problem. 1. Sums and differences For any set A of integers, we define the sumset A + A = {a + a : a, a ∈ A} and the difference set A − A = {a− a : a, a ∈ A}. In this section we consider finite sets of integers, and the relative sizes of their sumsets and difference sets. If A is a finite set of integers and x, y ∈ Z, then the translation of A by x is the set x + A = {x + a : a ∈ A} and the dilation of A by y is y ∗ A = {ya : a ∈ A}. We have (x + A) + (x + A) = 2x + 2A and (x + A) − (x + A) = A − A. Similarly, y ∗ A + y ∗ A = y ∗ (A + A) and y ∗ A − y ∗ A = y ∗ (A − A). It follows that |(x + y ∗ A) + (x + y ∗ A)| = |2A| and |(x + y ∗ A) − (x + y ∗ A)| = |A − A| so the cardinalities of the sum and difference sets of a finite set of integers are invariant under affine transformations of the set. 2000 Mathematics Subject Classification. 11B05, 11B13, 11B34.
Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science
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Citations in the corpus
- Some explicit constructions of sets with more sums than differences (2006, unknown): “Part One As before, let A ⊂ Z be a set of size 8, A = {ai : i = 1, ..., 8} where a1 > a2 > · · · > a8 = 0.”
- Sets characterized by missing sums and differences (2009, background): “We refer the reader to [7, 8] for the history of the problem.”
- Many sets have more sums than differences (2006, background): “N T ] 4 D ec 2 00 6 Many Sets Have More Sums Than Differences Greg Martin gerg@math.ubc.ca University of British Columbia and Kevin O’Bryant kevin@member.ams.org City University of New York, Staten Island March 30, 2022”