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Threshold functions for systems of equations on random sets
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Electronic Notes in Discrete Mathematics 43 (2013) 113–116 www.elsevier.com/locate/endm
Threshold functions for systems of equations on random sets Juanjo Ru´e 1,2 Instituto de Ciencias Matem´ aticas (CSIC-UAM-UCM-UC3M) Madrid, Spain
Ana Zumalac´arregui 3 Departamento de Matem´ aticas and Instituto de Ciencias Matem´ aticas Universidad Aut´ onoma de Madrid Madrid, Spain
Abstract We present a unified framework to deal with threshold functions for the existence of certain combinatorial structures in random sets. More precisely, let M · x = 0 be a linear system defining a fixed structure (k-arithmetic progressions, k-sums, Bh [g] sets or Hilbert cubes, for example), and A be a random set on 1, ..., n where each element is chosen independently with the same probability. We show that, under certain natural conditions, there exists a threshold function for the property “Am contains a non-trivial solution of M · x = 0” which only depends on the dimensions of M . We study the behavior of the limiting distribution of the number of non-trivial solutions in the threshold scale, and show that it follows a Poisson distribution in terms of volumes of certain convex polytopes arising from the linear system under study. Keywords: Threshold, Polytope, Sidon set, k-arithmetic progression
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.019
114
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
The existence of certain structures in large combinatorial systems plays a central role in discrete mathematics and, more specially, in combinatorial number theory. In the context of extremal combinatorics, this type of questions has provided an active area of research where many different techniques are used. In the general setting, finding extremal conditions is a difficult task and generally needs smart ad hoc arguments. One mayor example of this fact is the celebrated theorem of Szemer´edi on the existence of long arithmetic progressions in sets of positive density. Nevertheless, some results have been obtained by means of general arguments: in [5,6] upper and lower bounds for the size of maximal sets avoiding solutions to linear equations are obtained. We present a unified framework that includes many combinatorial structures and focus on the study of the common behavior of a random set in terms of the existence (or absence) of such structures. In this setting we can provide a clear picture of what is expected for most sets. This approach allows us to obtain results for a wide variety of structures via probabilistic methods, see [1]. The models of random sets we consider in this work are the analogues of the G(n, p) and G(n, M ) models in random graphs introduced by Erd˝os and R´enyi [2] in the 60’s. In the first case, for a probability p (depending possibly on n) we consider the random set A provided that P [a ∈ A] = p for every a ∈ [n]. In the former, we fix the number M of elements, and we n consider the uniform distribution among the M possible subsets of [n] with M elements. Despite the two models are not the same, they have similar . asymptotic behavior when choosing p = M n Let A ⊆ [n] be a random set, where every element is chosen with probability p, and M · x = 0 be a linear system of r equations in m variables. We study how the quantity |Am ∩ {x : M · x = 0}| behaves with respect to p and deduce the existence of a threshold function for the combinatorial property “Am contains a non-trivial solution of M · x = 0 ” in terms of the expected value for this random variable. 1
The first author is supported by a JAE-DOC grant from the JAE program in CSIC, Spain. The last author is supported by a FPU grant from Ministerio de Educacin, Ciencia y Deporte, Spain. Both authors were jointly financed by the MTM2011-22851 grant (Spain) and the ICMAT Severo Ochoa Project SEV-2011-0087 (Spain). 2 Email:[email protected] 3 Email:[email protected]
115
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
More precisely, we show that, under certain natural conditions for M , lim P
A
m
n→∞
contains a non-trivial
solution of M · x = 0
⎧ r 1− m ⎨ 0 if lim = 0, n→∞ pn = r 1− m ⎩ 1 if lim = ∞. n→∞ pn r
In other words, whenever the size of A is o(n m ) we can assure that asymptotically almost surely there are no other than trivial solutions of the linear system M · x = 0 with x ∈ Am . The main contribution in the study comes from those solutions whose components are pairwise distinct, since, roughly speaking, solutions with repeated components appear later in the regime. We also study the behavior of the limiting probability in the threshold scale. With this purpose, observe that the system of equations M · x = 0 and the restrictions on x define a non-empty, convex and rational polytope of dimension m − r, say PM . With this definition in mind, we show that there exists an exponential decay which depends on the volume of PM and the number of variables involved in the system of equations, but not on the r number of equations. More precisely, for p = cn m −1 , lim P
Am contains exactly t non-trivial
n→∞
solutions of M · x = 0
=
μt cmt −μcm e , t!
where μ is a constant which depends on the volume of the rational polytope PM and the symmetries of the system. r
Observe that, in particular, the previous result implies that for p = cn m −1 the number of solutions is Poisson distributed with parameter μcm , and, in particular lim P
n→∞
Am contains a non-trivial solution of M · x = 0
m
= 1 − e−μc .
We analize many interesting combinatorial families which fit into the presented scheme: arithmetic progression of length k, sum-free sets, Bh [g] or sets avoiding Hilbert cubes sets among others. In these examples it is clear what a trivial solution is, like an arithmetic progression x, x + d, x + 2d with d = 0. However, in the general setting there is no natural combinatorial definition. A key point on the general argument is to find the propper definition of trivial solution for a general system of equations that generalizes the concept of trivial in the previous examples and the one given in [5,6].
116
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
References [1] N. Alon and J.H Spencer, The probabilistic method, John Wiley & Sons Inc., 2008. [2] P. Erd˝os and A. R´enyi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ ozl., 5 (1960), 17–61. [3] A.P Godbole, S. Janson, N.W Locantore and R. Rapoport, Random Sidon sequences, J. Number Theory 75 1 (1999), 7–22. [4] J. Ru´e and A. Zumalacrregui, Threshold functions for systems of equations on random sets, arXiv:1212.5496. [5] I. Z. Ruzsa, Imre Z., Solving a linear equation in a set of integers. I, Acta Arith. 65 3 (1993), 259–282. [6] I. Z. Ruzsa, Solving a linear equation in a set of integers. II, Acta Arith. 72 4 (1995), 385–397. [7] C. S´andor, Non-degenerate Hilbert cubes in random sets , Journal de Th´eorie des Nombres de Bordeaux 19 1 (2007), 249–261.
Electronic Notes in Discrete Mathematics 43 (2013) 113–116 www.elsevier.com/locate/endm
Threshold functions for systems of equations on random sets Juanjo Ru´e 1,2 Instituto de Ciencias Matem´ aticas (CSIC-UAM-UCM-UC3M) Madrid, Spain
Ana Zumalac´arregui 3 Departamento de Matem´ aticas and Instituto de Ciencias Matem´ aticas Universidad Aut´ onoma de Madrid Madrid, Spain
Abstract We present a unified framework to deal with threshold functions for the existence of certain combinatorial structures in random sets. More precisely, let M · x = 0 be a linear system defining a fixed structure (k-arithmetic progressions, k-sums, Bh [g] sets or Hilbert cubes, for example), and A be a random set on 1, ..., n where each element is chosen independently with the same probability. We show that, under certain natural conditions, there exists a threshold function for the property “Am contains a non-trivial solution of M · x = 0” which only depends on the dimensions of M . We study the behavior of the limiting distribution of the number of non-trivial solutions in the threshold scale, and show that it follows a Poisson distribution in terms of volumes of certain convex polytopes arising from the linear system under study. Keywords: Threshold, Polytope, Sidon set, k-arithmetic progression
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.07.019
114
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
The existence of certain structures in large combinatorial systems plays a central role in discrete mathematics and, more specially, in combinatorial number theory. In the context of extremal combinatorics, this type of questions has provided an active area of research where many different techniques are used. In the general setting, finding extremal conditions is a difficult task and generally needs smart ad hoc arguments. One mayor example of this fact is the celebrated theorem of Szemer´edi on the existence of long arithmetic progressions in sets of positive density. Nevertheless, some results have been obtained by means of general arguments: in [5,6] upper and lower bounds for the size of maximal sets avoiding solutions to linear equations are obtained. We present a unified framework that includes many combinatorial structures and focus on the study of the common behavior of a random set in terms of the existence (or absence) of such structures. In this setting we can provide a clear picture of what is expected for most sets. This approach allows us to obtain results for a wide variety of structures via probabilistic methods, see [1]. The models of random sets we consider in this work are the analogues of the G(n, p) and G(n, M ) models in random graphs introduced by Erd˝os and R´enyi [2] in the 60’s. In the first case, for a probability p (depending possibly on n) we consider the random set A provided that P [a ∈ A] = p for every a ∈ [n]. In the former, we fix the number M of elements, and we n consider the uniform distribution among the M possible subsets of [n] with M elements. Despite the two models are not the same, they have similar . asymptotic behavior when choosing p = M n Let A ⊆ [n] be a random set, where every element is chosen with probability p, and M · x = 0 be a linear system of r equations in m variables. We study how the quantity |Am ∩ {x : M · x = 0}| behaves with respect to p and deduce the existence of a threshold function for the combinatorial property “Am contains a non-trivial solution of M · x = 0 ” in terms of the expected value for this random variable. 1
The first author is supported by a JAE-DOC grant from the JAE program in CSIC, Spain. The last author is supported by a FPU grant from Ministerio de Educacin, Ciencia y Deporte, Spain. Both authors were jointly financed by the MTM2011-22851 grant (Spain) and the ICMAT Severo Ochoa Project SEV-2011-0087 (Spain). 2 Email:[email protected] 3 Email:[email protected]
115
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
More precisely, we show that, under certain natural conditions for M , lim P
A
m
n→∞
contains a non-trivial
solution of M · x = 0
⎧ r 1− m ⎨ 0 if lim = 0, n→∞ pn = r 1− m ⎩ 1 if lim = ∞. n→∞ pn r
In other words, whenever the size of A is o(n m ) we can assure that asymptotically almost surely there are no other than trivial solutions of the linear system M · x = 0 with x ∈ Am . The main contribution in the study comes from those solutions whose components are pairwise distinct, since, roughly speaking, solutions with repeated components appear later in the regime. We also study the behavior of the limiting probability in the threshold scale. With this purpose, observe that the system of equations M · x = 0 and the restrictions on x define a non-empty, convex and rational polytope of dimension m − r, say PM . With this definition in mind, we show that there exists an exponential decay which depends on the volume of PM and the number of variables involved in the system of equations, but not on the r number of equations. More precisely, for p = cn m −1 , lim P
Am contains exactly t non-trivial
n→∞
solutions of M · x = 0
=
μt cmt −μcm e , t!
where μ is a constant which depends on the volume of the rational polytope PM and the symmetries of the system. r
Observe that, in particular, the previous result implies that for p = cn m −1 the number of solutions is Poisson distributed with parameter μcm , and, in particular lim P
n→∞
Am contains a non-trivial solution of M · x = 0
m
= 1 − e−μc .
We analize many interesting combinatorial families which fit into the presented scheme: arithmetic progression of length k, sum-free sets, Bh [g] or sets avoiding Hilbert cubes sets among others. In these examples it is clear what a trivial solution is, like an arithmetic progression x, x + d, x + 2d with d = 0. However, in the general setting there is no natural combinatorial definition. A key point on the general argument is to find the propper definition of trivial solution for a general system of equations that generalizes the concept of trivial in the previous examples and the one given in [5,6].
116
J. Rué, A. Zumalacárregui / Electronic Notes in Discrete Mathematics 43 (2013) 113–116
References [1] N. Alon and J.H Spencer, The probabilistic method, John Wiley & Sons Inc., 2008. [2] P. Erd˝os and A. R´enyi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ ozl., 5 (1960), 17–61. [3] A.P Godbole, S. Janson, N.W Locantore and R. Rapoport, Random Sidon sequences, J. Number Theory 75 1 (1999), 7–22. [4] J. Ru´e and A. Zumalacrregui, Threshold functions for systems of equations on random sets, arXiv:1212.5496. [5] I. Z. Ruzsa, Imre Z., Solving a linear equation in a set of integers. I, Acta Arith. 65 3 (1993), 259–282. [6] I. Z. Ruzsa, Solving a linear equation in a set of integers. II, Acta Arith. 72 4 (1995), 385–397. [7] C. S´andor, Non-degenerate Hilbert cubes in random sets , Journal de Th´eorie des Nombres de Bordeaux 19 1 (2007), 249–261.