Achievable sets in ℤn
-
Krishanu Roy Sankar
Abstract
What sets A ⊂ ℤn can be written in the form (K-K) ∩ ℤn, where K is a compact subset of ℝn such that K+ℤn = ℝn? Such sets A are called achievable, and it is known that if A is achievable, then 〈A〉 = ℤn. This condition completely characterizes achievable sets for n = 1, but not much is known for n ≥ 2. We attempt to characterize achievable sets further by showing that with any finite, symmetric set A ⊂ ℤn containing zero, we may associate a graph 𝒢(A). Then if A is achievable, we show that the set associated to some connected component of 𝒢(A) is achievable. In two dimensions, we can strengthen this theorem. Further generalizations and open questions are discussed. Throughout, the language and formalism of algebraic topology are useful.
Funding source: NSF
Award Identifier / Grant number: DMS 0754106
Funding source: NSA
Award Identifier / Grant number: H98230-09-1-0115
This research was done over the summer of 2010 at the University of Minnesota Duluth, with grants from the NSF (grant number DMS 0754106) and the NSA (grant number H98230-09-1-0115). I would like to thank Joe Gallian for running the program and for his advice during my research. I would also like to thank my advisors, Nathan Pflueger and Maria Monks, for their help in my research. I especially would like to thank Nathan Pflueger and Joe Gallian for their help with this paper.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- On the character degrees of Sylow p-subgroups of Chevalley groups G(pf) of type E
- Achievable sets in ℤn
- A criterion on oscillation and variation for the commutators of singular integral operators
- Some 𝐇1𝔉-groups with unbounded torsion and a conjecture of Kropholler and Mislin
- Goodwillie calculus and Whitehead products
- Computing Borel's regulator
- Applications of a transcendence criterion for reciprocal sums of binary linear recurrences
- BSDE and generalized Dirichlet forms: The infinite dimensional case
- Faithful representations of infinite-dimensional nilpotent Lie algebras
- Locally finitely presented categories with no flat objects
- A common generalization of metric, ultrametric and topological fixed point theorems
- Correction to A common generalization of metric, ultrametric and topological fixed point theorems [Forum Math. 27 (2015), 303–327]
- Erdős–Rényi graph, Szemerédi–Trotter type theorem, and sum-product estimates over finite rings
- Minimal support results for Schrödinger equations
- Dynamics for the focusing, energy-critical nonlinear Hartree equation
- A fixed point theorem for Lie groups acting on buildings and applications to Kac–Moody theory
- On effective determination of Maass forms from central values of Rankin–Selberg L-function
- On embedding left-ordered groups into division rings
- Conservation property of symmetric jump-diffusion processes
- Irreducible representations of Leavitt path algebras
- Commutators of elements of coprime orders in finite groups
- Endoscopy and the transfer from GSp(4) to GL(4)
- Cycles in Leavitt path algebras by means of idempotents
- On volumes determined by subsets of Euclidean space
Articles in the same Issue
- Frontmatter
- On the character degrees of Sylow p-subgroups of Chevalley groups G(pf) of type E
- Achievable sets in ℤn
- A criterion on oscillation and variation for the commutators of singular integral operators
- Some 𝐇1𝔉-groups with unbounded torsion and a conjecture of Kropholler and Mislin
- Goodwillie calculus and Whitehead products
- Computing Borel's regulator
- Applications of a transcendence criterion for reciprocal sums of binary linear recurrences
- BSDE and generalized Dirichlet forms: The infinite dimensional case
- Faithful representations of infinite-dimensional nilpotent Lie algebras
- Locally finitely presented categories with no flat objects
- A common generalization of metric, ultrametric and topological fixed point theorems
- Correction to A common generalization of metric, ultrametric and topological fixed point theorems [Forum Math. 27 (2015), 303–327]
- Erdős–Rényi graph, Szemerédi–Trotter type theorem, and sum-product estimates over finite rings
- Minimal support results for Schrödinger equations
- Dynamics for the focusing, energy-critical nonlinear Hartree equation
- A fixed point theorem for Lie groups acting on buildings and applications to Kac–Moody theory
- On effective determination of Maass forms from central values of Rankin–Selberg L-function
- On embedding left-ordered groups into division rings
- Conservation property of symmetric jump-diffusion processes
- Irreducible representations of Leavitt path algebras
- Commutators of elements of coprime orders in finite groups
- Endoscopy and the transfer from GSp(4) to GL(4)
- Cycles in Leavitt path algebras by means of idempotents
- On volumes determined by subsets of Euclidean space
