Upper tails for arithmetic progressions in random subsets

Warnke, Lutz

doi:10.1007/s11856-017-1546-3

Cited by 38 publications

(107 citation statements)

References 50 publications

(220 reference statements)

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“…Related results for the upper tail of arithmetic progressions and Schur triples have been established by Warnke [30].…”

Section: Random Subsets Of the Integersmentioning

confidence: 88%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

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The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability P(X (1 − ε)EX) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ε = 1) of Janson, Luczak and Ruciński.

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“…Related results for the upper tail of arithmetic progressions and Schur triples have been established by Warnke [30].…”

Section: Random Subsets Of the Integersmentioning

confidence: 88%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

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“…= O(1), because the validity condition (27) implies s rj +1 ≤ s rj , since t rj = 0 . (b) If 2 ≤ t rj +1 ≤ s rj +1 − 1, then we have g t,s (r j )g t,s (r j + 1) σ sr j σ sr j +1 = n 2 s −1 rj p sr j · s 2 rj s 2 rj+1 p sr j +1−tr j +1 · σ −1 sr j σ −1 sr j +1…”

Section: 2mentioning

confidence: 99%

Bivariate fluctuations for the number of arithmetic progressions in random sets

Barhoumi-Andréani¹,

Koch²,

Liu³

2019

Electron. J. Probab.

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We study arithmetic progressions {a, a+b, a+2b, . . . , a+(ℓ−1)b}, with ℓ ≥ 3, in random subsets of the initial segment of natural numbers [n] := {1, 2, . . . , n}. Given p ∈ [0, 1] we denote by [n]p the random subset of [n] which includes every number with probability p, independently of one another. The focus lies on sparse random subsets, i.e. when p = p(n) = o(1) as n → +∞.Let X ℓ denote the number of distinct arithmetic progressions of length ℓ which are contained in [n]p. We determine the limiting distribution for X ℓ not only for fixed ℓ ≥ 3 but also when ℓ = ℓ(n) → +∞. The main result concerns the joint distribution of the pair (X ℓ , X ℓ ′ ), ℓ > ℓ ′ , for which we prove a bivariate central limit theorem for a wide range of p. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or nontrivial is characterised by the asymptotic behaviour (as n → +∞) of the threshold function ψ ℓ = ψ ℓ (n) := np ℓ−1 ℓ. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

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“…We will, however, not work with it in its entirety; instead of breaking the aforementioned “

normallog false(n false) false/ normallog normallog false(n false)

barrier”, we will work around it. Our method is inspired by the idea of r‐star matchings described in Section 3.2 of Warnke .…”

Section: Concentration (Step Iii)mentioning

confidence: 99%

“…Next, fix somẽ> 0 and define P 21 ∶= { n ∈ P 2 ∶ A(n) ≤ n 1∕(h−1) L(n) −̃} and P 22 ∶= P 2 ⧵ P 21 .…”

Section: Lemma 54mentioning

confidence: 99%

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An extension of the Erdős‐Tetali theorem

Táfula

2018

Random Struct Algorithms

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Given a sequence 𝒜=false{a00, then there must exist 𝒜⊆double-struckN with false|𝒜∩false[0,xfalse]false|=normalΘfalse(ffalse(xfalse)false) for which r𝒜,h + ℓ(n) = Θ(f(n)h + ℓ/n) for all ℓ ≥ 0. Furthermore, for h = 2 this condition can be weakened to x1false/2normallogfalse(xfalse)1false/2≪ffalse(xfalse)≪x. The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.

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Upper tails for arithmetic progressions in random subsets

Cited by 38 publications

References 50 publications

The Lower Tail: Poisson Approximation Revisited

The Lower Tail: Poisson Approximation Revisited

Bivariate fluctuations for the number of arithmetic progressions in random sets

An extension of the Erdős‐Tetali theorem

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