Extremal problems about asymptotic bases: a survey

Grekos, Georges

doi:10.1515/9783110925098.237

Cited by 5 publications

(10 citation statements)

References 7 publications

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“…It is conjectured by Erdős and Graham [4] that there is a constant α such that X(h) ∼ αh 2 as h → ∞, but this remains open. The inequalities in (5) imply that X(1) = 1, X(2) = 4, X(3) = 7, but even the value of X(4) remains unknown.…”

Section: Introductionmentioning

confidence: 99%

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Additive bases in groups

2017

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Section: Introductionmentioning

confidence: 99%

“…In [6,7], Grekos observed that in examples of bases A of order h that give large values of X(h), there are actually very few elements a ∈ A such that ord * (A \ {a}) = X(h). This led him to introduce the function…”

Section: Introductionmentioning

confidence: 99%

Additive bases in groups

2017

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“…This seems far from being sufficient to conjecture that S(h) = h + 1 (a formula false for h = 1) rather than S(h) = 2h − 1 (for instance). Also, we may notice that the upper estimate in (1.5) gives a new proof of S(3) ≤ 6, an upper bound first proved in [8]. It is still undecided whether S(3) is 4, 5 or 6.…”

Section: Theorem 2 For Any Positive Integer H S(h) ≤ 2hmentioning

confidence: 96%

“…Motivated by this conjecture, Grekos himself proved that S(3) ≤ 6 < X(3) (see [8]) and announced, as a consequence of [6], that S(2) = 3 (actually, in [6], this result is proved under an additional minimality condition). But Grekos' proof was never published.…”

Section: Introductionmentioning

confidence: 98%

Grekos’ S function has a linear growth

Cassaigne

Plagne

2004

Proc. Amer. Math. Soc.

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Abstract. An exact additive asymptotic basis is a set of nonnegative integers such that there exists an integer h with the property that any sufficiently large integer can be written as a sum of exactly h elements of A. The minimal such h is the exact order of A (denoted by ord * (A)). Given any exact additive asymptotic basis A, we define A * to be the subset of A composed with the elements a ∈ A such that A \ {a} is still an exact additive asymptotic basis. It is known that A \ A * is finite.In this framework, a central quantity introduced by Grekos is the function S(h) defined as the following maximum (taken over all bases A of exact order h):In this paper, we introduce a new and simple method for the study of this function. We obtain a new estimate from above for S which improves drastically and in any case on all previously known estimates. Our estimate, namely S(h) ≤ 2h, cannot be too far from the truth since S verifies S(h) ≥ h + 1. However, it is certainly not always optimal since S(2) = 3. Our last result shows that S(h) is in fact a strictly increasing sequence.

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“…Unfortunately, in the general case, the best that is known is (h 2), h+1 S(h) X(h), the lower bound being due to Ha rtter [7]. Grekos' conjecture is that, for h>1, S(h)<X(h), a conjecture supported by S(2)=3<X(2) and S(3) 6<X(3) (proved in [3,5], respectively).…”

Section: Introductionmentioning

confidence: 99%