Discrete Kakeya-type problems and small bases

Alon, Noga; Bukh, Boris; Sudakov, Benny

doi:10.1007/s11856-009-0115-9

Cited by 5 publications

(9 citation statements)

References 15 publications

(14 reference statements)

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“…As remarked above, for r large the bound of Theorem 4 (and consequently, that of Corollary 5) is rather weak. The best possible construction we can give in this regime does not take linearity into account and is just a universal set construction where, following [ABS09], we say that a subset of a group is k-universal if it contains a translate of every k-element subset of the group. As shown in [ABS09], every finite abelian group G possesses a k-universal subset of size at most 8 k−1 k|G| 1−1/k .…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…The best possible construction we can give in this regime does not take linearity into account and is just a universal set construction where, following [ABS09], we say that a subset of a group is k-universal if it contains a translate of every k-element subset of the group. As shown in [ABS09], every finite abelian group G possesses a k-universal subset of size at most 8 k−1 k|G| 1−1/k . In our present context the group under consideration is the additive group of the vector space F n q , in which case we were able to give a particularly simple construction of universal sets and refine slightly the bound just mentioned.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

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Kakeya-type sets in finite vector spaces

Kopparty¹,

Lev

Saraf³

et al. 2011

J Algebr Comb

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Abstract. For a finite vector space V and a non-negative integer r ≤ dim V we estimate the smallest possible size of a subset of V , containing a translate of every rdimensional subspace. In particular, we show that if K ⊆ V is the smallest subset with this property, n denotes the dimension of V , and q is the size of the underlying field, then for r bounded and r < n ≤ rq r−1 we have |V \ K| = Θ(nq n−r+1 ); this improves previously known bounds |V \ K| = Ω(q n−r+1 ) and |V \ K| = O(n 2 q n−r+1 ).

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Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Kakeya-type sets in finite vector spaces

Kopparty¹,

Lev

Saraf³

et al. 2011

J Algebr Comb

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“…Similarly, if m is at least n 1/2+ then almost all m-element sets require a basis of size at least c √ n. For the borderline case when m is of the order √ n their counting argument only yields existence of sets that need a basis of size c √ n log log n/ log n, and they asked if every m-set of size m = √ n has a basis with o(m) elements. This is established in [7], where it is shown that in fact any such set has a basis of size O( √ n log log n/ log n). The argument is probabilistic.…”

Section: Additive Number Theorymentioning

confidence: 90%

“…. , n} is of size Ω(n 1− ) for every > 0 and d ≥ 2, only a modest improvement of the n 2/3−o(1) lower bound of Erdős and Newman for large values of d is proved in [7], where it is shown that the set {t d : t = 1, . .…”

Section: Additive Number Theorymentioning

confidence: 99%

Paul Erdős and Probabilistic Reasoning

Alon

2013

Bolyai Society Mathematical Studies

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One of the major contributions of Paul Erdős is the development of the Probabilistic Method and its applications in Combinatorics, Graph Theory, Additive Number Theory and Combinatorial Geometry. This short paper describes some of the beautiful applications of the method, focusing on the long-term impact of the work, questions and results of Erdős. This is mostly a survey, but it contains a few novel results as well. The Probabilistic MethodThe Probabilistic Method is one of the most significant contributions of Paul Erdős, and part of his greatness is the fact that applications of the probabilistic method and of random graphs have become so common that it is now possible to use those without explicitly mentioning him. The method is a powerful tool with numerous applications in Combinatorics, Graph theory, Additive Number Theory and Geometry and had an immense impact on the development of theoretical Computer Science as well. The results and tools are far too numerous to cover in a short survey, even if the focus is only on those influenced directly by the work and problems of Erdős, and thus this paper is mainly a selection of topics that illustrate the method, and is not meant to be a comprehensive treatment of the whole area. Several books that contain more material on the subject are [13], [18], [54], [61].It is convenient to classify the applications of probabilistic techniques in Discrete Mathematics into three groups. The first one deals with the study of random combinatorial objects, like random graphs or random matrices. The results here are essentially results in Probability Theory, although many of them are motivated by problems in Combinatorics. The second group consists of probabilistic constructions. These are applications of probabilistic arguments in order to prove the existence of combinatorial structures which satisfy a list of prescribed properties. Existence proofs of this type often supply extremal examples to various questions in Discrete Mathematics. The third group, which contains some of the most striking examples, focuses on the application of probabilistic reasoning in the proofs of deterministic statements whose formulation does not give any indication that randomness may be helpful in their study.

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“…Certainly, multiplicative energy E × (A, B) can be expressed in terms of multiplicative convolutions, similar to (1). Usually we will use the additive energy and write E(A, B) instead of E + (A, B).…”

Section: Notationmentioning

confidence: 99%

An introduction to higher energies and sumsets

Shkredov

2015

Preprint

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These notes basically contain a material of two mini-courses which were read in Göteborg in April 2015 during the author visit of Chalmers & Göteborg universities and in Beijing in November 2015 during "Chinese-Russian Workshop on Exponential Sums and Sumsets". The article is a short introduction to a new area of Additive Combinatorics which is connected which so-called the higher sumsets as well as with the higher energies. We hope the notes will be helpful for a reader who is interested in the field.

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Discrete Kakeya-type problems and small bases

Cited by 5 publications

References 15 publications

Kakeya-type sets in finite vector spaces

Kakeya-type sets in finite vector spaces

Paul Erdős and Probabilistic Reasoning

An introduction to higher energies and sumsets

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