Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov

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Displaying similar documents to “Product sets cannot contain long arithmetic progressions”

Herbrand consistency and bounded arithmetic

Zofia Adamowicz (2002)

Fundamenta Mathematicae

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We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove $T ₘ ⊬ H C o n s^{I ₘ} (T ₘ)$ , where $H C o n s^{I ₘ}$ is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.

О моделях арифметики в $A$ -изолиях.

В.Л. Михеев (1991)

Algebra i Logika

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A note on Δ₁ induction and Σ₁ collection

Neil Thapen (2005)

Fundamenta Mathematicae

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Slaman recently proved that Σₙ collection is provable from Δₙ induction plus exponentiation, partially answering a question of Paris. We give a new version of this proof for the case n = 1, which only requires the following very weak form of exponentiation: " $x^{y}$ exists for some y sufficiently large that x is smaller than some primitive recursive function of y".

Riesz sequences and arithmetic progressions

Itay Londner, Alexander Olevskiĭ (2014)

Studia Mathematica

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Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E (Λ) : = e_{λ \in Λ}^{i λ t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $ℓ = O (N^{1 - ε})$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that...

On integers of the form $p + 2^{k}$

Laurent Habsieger, Xavier-François Roblot (2006)

Acta Arithmetica

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Automorphisms of models of bounded arithmetic

Ali Enayat (2006)

Fundamenta Mathematicae

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We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $= I_{f i x} (j)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here...

Fonction de Seshadri arithmétique en géométrie d’Arakelov

Huayi Chen (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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To any adelic invertible sheaf on a projective arithmetic variety and any regular algebraic point of the arithmetic variety, we associate a function defined on $ℝ$ which measures the separation of jets on this algebraic point by the “small” sections of the adelic invertible sheaf. This function will be used to study the arithmetic local positivity.

On the lonely runner conjecture

Ram Krishna Pandey (2010)

Mathematica Bohemica

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Suppose $k + 1$ runners having nonzero distinct constant speeds run laps on a unit-length circular track. The Lonely Runner Conjecture states that there is a time at which a given runner is at distance at least $1 / (k + 1)$ from all the others. The conjecture has been already settled up to seven ( $k \leq 6$ ) runners while it is open for eight or more runners. In this paper the conjecture has been verified for four or more runners having some particular speeds using elementary tools.

On the structure of sets with many k-term arithmetic progressions

(2009)

Acta Arithmetica

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Aposyndesis in $ℕ$

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2023)

Commentationes Mathematicae Universitatis Carolinae

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We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P (a, b)$ with the property that every prime number that divides $a$ also divides $b$ , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are...

A generalization of Freiman's 3k-3 Theorem

Yahya Ould Hamidoune, Alain Plagne (2002)

Acta Arithmetica

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On the weak pigeonhole principle

Jan Krajíček (2001)

Fundamenta Mathematicae

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We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs of the Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions...

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Some results on Oppenheim's "Factorisatio Numerorum" function

Florian Luca, Anirban Mukhopadhyay, Kotyada Srinivas (2010)

Acta Arithmetica

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Some proofs of undecidability of arithmetic

Andrzej Grzegorczyk (1956)

Fundamenta Mathematicae

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Displaying similar documents to “Product sets cannot contain long arithmetic progressions”

Herbrand consistency and bounded arithmetic

О моделях арифметики в $A$ -изолиях.

A note on Δ₁ induction and Σ₁ collection

Riesz sequences and arithmetic progressions

On integers of the form $p + 2^{k}$

Automorphisms of models of bounded arithmetic

Fonction de Seshadri arithmétique en géométrie d’Arakelov

On the lonely runner conjecture

On the structure of sets with many k-term arithmetic progressions

Aposyndesis in $ℕ$

A generalization of Freiman's 3k-3 Theorem

On the weak pigeonhole principle

Algebra with Arithmetic and Mensuration

The Elements of Arithmetic

On disjoint arithmetic progressions

Some results on Oppenheim's "Factorisatio Numerorum" function

Some proofs of undecidability of arithmetic