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On volumes determined by subsets of Euclidean space
-
Allan Greenleaf
,
Alex Iosevich
Veröffentlicht/Copyright:
19. Juni 2013
Abstract
Given E ⊂ ℝd, define the volume set of E, 𝒱(E) = {det(x1,x2,...,xd) : xj ∈ E}. In ℝ3, we prove that 𝒱(E) has positive Lebesgue measure if either the Hausdorff dimension of E ⊂ ℝ3 is greater than 13/5, or E is a product set of the form E = B1×B2×B3 with Bj ⊂ ℝ, dimℋ(Bj) > 2/3, j = 1,2,3. We show that the same conclusion holds for 𝒱(E) of Salem subsets E ⊂ ℝd with dimℋ(E) > d - 1, and give applications to discrete combinatorial geometry.
MSC: 42B35
Funding source: NSF
Award Identifier / Grant number: DMS-0853892
Funding source: NSF
Award Identifier / Grant number: DMS-1045404
Funding source: Fondation de Mathématiques Jacques Hadamard (FMJH)
Award Identifier / Grant number: Sophie Germain International post-doctoral scholarship
Received: 2011-11-1
Revised: 2012-9-16
Published Online: 2013-6-19
Published in Print: 2015-1-1
© 2015 by De Gruyter
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Artikel in diesem Heft
- 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
Schlagwörter für diesen Artikel
Bilinear operators;
Falconer distance problem;
finite point configurations
Artikel in diesem Heft
- 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
