scholarly journals Thin set theorems and cone avoidance

2020 ◽  
Vol 373 (4) ◽  
pp. 2743-2773
Author(s):  
Peter Cholak ◽  
Ludovic Patey
Keyword(s):  
Thin Set

scholarly journals Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

Self-Similarity Bounds for Locally Thin Set Families

2001 ◽  
Vol 10 (4) ◽  
pp. 309-315
Author(s):  
EMANUELA FACHINI ◽  
JÁNOS KÖRNER ◽  
ANGELO MONTI
Keyword(s):  
Upper Bound ◽  
Maximum Size ◽  
Self Similarity ◽  
Thin Set

A family of subsets of an n-set is k-locally thin if, for every k-tuple of its members, the ground set has at least one element contained in exactly one of them. For k = 5 we derive a new exponential upper bound for the maximum size of these families. This implies the same bound for all odd values of k > 3. Our proof uses the graph entropy bounding technique to exploit a self-similarity in the structure of the hypergraph associated with such set families.

scholarly journals On the relationship between ordinary thin sets and full-thin sets

1980 ◽  
Vol 29 (3) ◽  
pp. 348-361
Author(s):  
J. S. Hwang ◽  
H. L. Jackson
Keyword(s):  
Unit Ball ◽  
Half Space ◽  
Boundary Point ◽  
Subject Classification ◽  
Thin Set ◽  
Thin Sets ◽  
The Relationship

AbstractIn this work we demonstrate that if Ω ⊂ Rn (n ≧ 3) is either a half space or a unit ball, and if E ⊂ ω then E is an ordinary thin set at a boundary point of Ω (including the point at infinity if Ω is a half space) if and only if it is a full-thin set at the corresponding Kuramochi boundary point of Ω. The case for n = 2 has already been considered in an earlier work.1980 Mathematics subject classification (Amer. Math. Soc.): 31 B 05.

A Thin Set of Circles

1968 ◽  
Vol 75 (10) ◽  
pp. 1077-1081 ◽  
Author(s):  
J. R. Kinney
Keyword(s):  
Thin Set

Beurling-Dahlberg-Sjögren Type Theorems for Minimally Thin Sets in a Cone

2003 ◽  
Vol 46 (2) ◽  
pp. 252-264 ◽  
Author(s):  
Ikuko Miyamoto ◽  
Minoru Yanagishita ◽  
Hidenobu Yoshida
Keyword(s):  
Half Space ◽  
Corner Point ◽  
Smooth Boundary ◽  
Thin Set ◽  
Special Domain ◽  
Thin Sets ◽  
Minimally Thin Set

AbstractThis paper shows that some characterizations of minimally thin sets connected with a domain having smooth boundary and a half-space in particular also hold for the minimally thin sets at a corner point of a special domain with corners, i.e., the minimally thin set at ∞ of a cone.

A Thin Set of Circles

1968 ◽  
Vol 75 (10) ◽  
pp. 1077 ◽  
Author(s):  
J. R. Kinney
Keyword(s):  
Thin Set

scholarly journals Constructing sequences one step at a time

2020 ◽  
Vol 20 (03) ◽  
pp. 2050017
Author(s):  
Henry Towsner
Keyword(s):  
Reverse Mathematics ◽  
New Method ◽  
Well Quasi Order ◽  
Thin Set ◽  
One Step ◽  
Quasi Order ◽  
Weak König’S Lemma ◽  
König’S Lemma

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain ([Formula: see text]) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma ([Formula: see text]), including showing that [Formula: see text] does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between [Formula: see text] and the Ascending/Descending Sequences principle, even in the presence of [Formula: see text].

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