Sum-Free Sets

1972 ◽  
pp. 124-271 ◽  
Author(s):  
Anne Penfold Street
Keyword(s):  
Free Sets

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

On the Density of Universal Sum-Free Sets

1999 ◽  
Vol 8 (3) ◽  
pp. 277-280 ◽  
Author(s):  
TOMASZ SCHOEN
Keyword(s):  
Free Set ◽  
Free Sets

A set A is called universal sum-free if, for every finite 0–1 sequence χ = (e1, …, en), either(i) there exist i, j, where 1[les ]j<i[les ]n, such that ei = ej = 1 and i − j∈A, or(ii) there exists t∈N such that, for 1[les ]i[les ]n, we have t + i∈A if and only if ei = 1.It is proved that the density of each universal sum-free set is zero, which settles a problem of Cameron.

Sum-free sets and short covering codes

2010 ◽  
Vol 39 (6) ◽  
Author(s):  
Emerson Carmelo ◽  
Candido Mendonça
Keyword(s):  
Covering Codes ◽  
Short Covering ◽  
Free Sets

scholarly journals Lower bounds for corner-free sets

10.53733/86 ◽  
2021 ◽  
Vol 51 ◽  
pp. 1-2
Author(s):  
Ben Green
Keyword(s):  
Lower Bounds ◽  
Free Sets

We show that for infinitely many $N$ there is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not containing any configuration $(x, y), (x + d, y), (x, y + d)$ with $d \neq 0$, where $c = 2 \sqrt{2 \log_2 \frac{4}{3}} \approx 1.822\dots$.

scholarly journals On sets with small sumset and m-sum-free sets in Z/pZ

2021 ◽  
Vol 149 (1) ◽  
pp. 155-177
Author(s):  
Pablo Candela ◽  
David Gonzalez-Sanchez ◽  
David Grynkiewicz
Keyword(s):  
Free Sets

Generalized heredity in ℬ-free systems

2021 ◽  
pp. 2140008
Author(s):  
Gerhard Keller
Keyword(s):  
Orbit Closure ◽  
Light Tails ◽  
Free Sets

Let [Formula: see text] be a primitive set, [Formula: see text], [Formula: see text], and denote by [Formula: see text] the orbit closure of [Formula: see text] under the shift. We complement results on heredity of [Formula: see text] from [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489] in two directions: In the proximal case we prove that a certain subshift [Formula: see text], which coincides with [Formula: see text] when [Formula: see text] is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set [Formula: see text] has light tails, as in [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that [Formula: see text] is always “hereditary above its unique minimal (Toeplitz) subsystem”. Finally, we characterize this Toeplitz subsystem as being a set [Formula: see text], where [Formula: see text] for a set [Formula: see text] that can be derived from [Formula: see text], and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of [Formula: see text]-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690–2734] are heavily used.

On the number of sum-free sets in a segment of positive integers

2002 ◽  
Vol 12 (4) ◽  
Author(s):  
K. G. Omelyanov ◽  
A. A. Sapozhenko
Keyword(s):  
Basic Research ◽  
Basic Research Grant ◽  
Positive Integers ◽  
Free Sets ◽  
Research Grant

AbstractA set A of integers is called sum-free if a + b ∉ A for any a, b ∈ A. For an arbitrary Ɛ > 0, let ssThis research was supported by the Russian Foundation for Basic Research, grant 01-01-00266.

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