Threshold functions and Poisson convergence for systems of equations in random sets

Abstract

Abstract. We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets, B h [g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "A contains a non-trivial solution of M ·x = 0" where A is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1, . . . , n}. Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation.Furthermore, we study the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.

A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

“…Note that we need to cover the whole range of 0 ≤ p(n) ≤ c n −1/m1(A) since we are not dealing with a monotone property. Markov's Inequality and Lemma 2.7 therefore give us lim n→∞ P | S 0 (A) ∩ [n] m p | = 0 = 0, see also Rué et al [14]. It clearly follows that we also have lim n→∞ P ([n] p → ǫ A) = 0 for any ǫ > 0 if p = p(n) ≪ n −1/m(A) .…”

“…Lastly, A = ( 1 1 −r ) for r ∈ N\{1, 2} is neither partition nor density regular but it is abundant. For some more examples see [14].…”

“…In order to determine the exact asymptotic value of | S(A)∩[n] m |/n m−rk(A) or | S 0 (A)∩[n] m |/n m−rk(A) , one needs to employ Ehrhart's Theory, see for example Rué et al [14].…”

“…For singe-line equations it is common in combinatorial number theory to not just limit oneself to proper solutions, that is solutions with no repeated entries, but to also consider certain non-trivial solutions which may have some repeated entries. Ruzsa [13] gave a definition for non-trivial solutions in this scenario and more recently Rué, S. and Zumalacárregui [14] extended it to include arbitrary homogeneous linear systems of equations. A third and final goal will therefore be to extend the previously stated sparse results to include non-trivial solutions.…”

On the optimality of the uniform random strategy

“…It follows that Theorem 1.3 applies and due to (19) establishes the desired upper bound on the threshold bias. ▪…”