Shattering News

Anstee, Richard P.; Rónyai, Lajos; Sali, Attila

doi:10.1007/s003730200003

Cited by 43 publications

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“…, n} (a so-called set system) by considering a word w ∈ {0, 1} n as the characteristic function of a subset. Proposition 2.3 has been proven in relation to set systems in [3]…”

Section: Word Gamesmentioning

confidence: 93%

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On winning shifts of marked uniform substitutions

Peltomäki

Salo

2019

RAIRO-Theor. Inf. Appl.

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The second author introduced with I. Törmä a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131-152]. The game has a predetermined (possibly finite) choice sequence α 1 , α 2 , . . . of integers such that on round n the player A chooses a subset S n of size α n of some fixed finite alphabet and the player B picks a letter from the set S n . The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that W(X) and X have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.

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“…, n} (a so-called set system) by considering a word w ∈ {0, 1} n as the characteristic function of a subset. Proposition 2.3 has been proven in relation to set systems in [3]…”

Section: Word Gamesmentioning

confidence: 93%

“…In Table 1, we list irreducible choice sequences of W(τ) for lengths 1 to 24. 3 For the choice sequence 2212, Alice has the following winning strategy:…”

Section: The Motivating Example Of the Thue-morse Substitutionmentioning

confidence: 99%

On winning shifts of marked uniform substitutions

Peltomäki

Salo

2019

RAIRO-Theor. Inf. Appl.

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“…, n} of size at most m − 1 equals m−1 k=0 n k . This stronger statement is due to Pajor [Paj85], and the resulting very short inductive proof which we shall now sketch for completeness appears as Theorem 1.1 in [ARS02].…”

Section: Sauer-shelah the Sauer-shelah Lemma [Sau72 She72mentioning

confidence: 94%

Restricted Invertibility Revisited

Youssef

2017

A Journey Through Discrete Mathematics

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Suppose that m, n ∈ N and that A : R m → R n is a linear operator. It is shown here that if k, r ∈ N satisfy k < r rank(A) then there exists a subset σ ⊆ {1, . . . , m} with |σ| = k such that the restriction of A to R σ ⊆ R m is invertible, and moreover the operator norm of the inverse. . . sm(A) are the singular values of A. This improves over a series of works, starting from the seminal Bourgain-Tzafriri Restricted Invertibility Principle, through the works of Vershynin, Spielman-Srivastava and Marcus-Spielman-Srivastava. In particular, this directly implies an improved restricted invertibility principle in terms of Schatten-von Neumann norms.

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“…VC dimension (and the attendant Sauer-Shelah-Perles lemma) has been extended to various settings, such as non-binary vectors [1,16,18,30], integer vectors [33], Boolean matrices with forbidden configurations [3,4], multivalued functions [16], continuous spaces [25], graph powers [8], and ordered variants [5]. In this paper, we formulate a new generalization of VC dimension, to graded lattices, and prove a Sauer-Shelah-Perles lemma for lattices with nonvanishing Möbius function, a rich class of lattices which includes the lattice of subspaces of a finite vector space as well as all geometric lattices (flats of matroids).…”

Section: Introductionmentioning

confidence: 99%

A Sauer-Shelah-Perles Lemma for Sumsets

Dvir¹,

Moran

2018

Electron. J. Combin.

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We extend the Sauer-Shelah-Perles lemma to an abstract setting that is formalized using the language of lattices. Our extension applies to all finite lattices with nonvanishing Möbius function, a rich class of lattices which includes all geometric lattices (or matroids) as a special case.For example, our extension implies the following result in Algebraic Combinatorics: let F be a family of subspaces of F n q . We say that F shatters a subspace U if for every subspace U ′ ≤ U there is F ∈ F such that F ∩U = U ′ . Then, if |F| > n 0 q +· · ·+ n d q then F shatters some (d+1)-dimensional subspace (where n k q denotes the number of k-dimensional subspaces in

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Shattering News

Cited by 43 publications

References 0 publications

On winning shifts of marked uniform substitutions

On winning shifts of marked uniform substitutions

Restricted Invertibility Revisited

A Sauer-Shelah-Perles Lemma for Sumsets

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