Topic
Thin set
About: Thin set is a research topic. Over the lifetime, 46 publications have been published within this topic receiving 601 citations.
Papers
TL;DR: Weihrauch reducibility as mentioned in this paper has been studied in the context of combinatorial problems, and it has been used to compare and contrast with the traditional notion of implication in reverse mathematics.
Abstract: The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n; j; k 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely ner metric by which to gauge the relative strength of mathematical propositions. We also study Weak K�onig's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve in nitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of di erence between combinatorial problems previously thought to be more closely related.
110 citations
Posted Content•
TL;DR: The notion of uniform reducibility has been studied in the context of combinatorial problems as discussed by the authors, and it has been used to compare and contrast with the traditional notion of implication in reverse mathematics.
Abstract: The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to non-uniform decisions about how to proceed in a given construction. In practice, however, if a theorem $\mathsf{Q}$ implies a theorem $\mathsf{P}$, it is usually because there is a direct uniform translation of the problems represented by $\mathsf{P}$ into the problems represented by $\mathsf{Q}$, in a precise sense. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all $n,j,k \geq 1$, if $j < k$ then Ramsey's theorem for $n$-tuples and $k$ many colors does not uniformly reduce to Ramsey's theorem for $j$ many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak Konig's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Uniform reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.
54 citations
TL;DR: In this paper, it was shown that every sufficiently large odd integer can be written as the sum of three primes when the primes so utilized are restricted to an explicit thin set.
Abstract: It was proved by Vinogradov that every sufficiently large odd integer can be written as the sum of three primes. We show that this remains the case when the primes so utilized are restricted to an explicit thin set. One may take, for example, the «Piatetski-Shapiro primes» p = [n 1/γ ] with any γ>20/21. By a similar argument it would follow that, for arbitrary θ, 0 0, one may take the set of primes for which {p θ }
49 citations
TL;DR: In this paper, the exceptional set in Manin's conjecture is shown to be a thin set using the minimal model program and boundedness of log Fano varieties, and it is shown that it is bounded by a constant number of rational points of a bounded height.
Abstract: Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.
41 citations
TL;DR: In this paper, it was shown that if f : X → S is a dominant morphism of K-varieties and both S and all fibers f −1 (s), s ∈ S(K), are of Hilbert type, then so is X.
Abstract: A variety X over a field K is of Hilbert type if X(K) is not thin. We prove that if f : X → S is a dominant morphism of K-varieties and both S and all fibers f −1 (s), s ∈ S(K), are of Hilbert type, then so is X. We apply this to answer a question of Serre on products of varieties and to generalize
21 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 1 |
| 2019 | 2 |
| 2018 | 3 |
| 2017 | 1 |
| 2016 | 4 |