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Let $A subset mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($vert NAvert = P_A(N)$ for some $P_A(X) in mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional sets), once $N$ is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary $A$. Such explicit results were only previously known in the special cases when $d=1$, when the convex hull of $A$ is a simplex or when $vert Avert = d+2$, results which we improve.
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When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the sets underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in finite fields of prime order. Finally, for sufficiently small finite fields we computationally calculate the minimum for all prime orders.
We determine the asymptotics of the number of independent sets of size $lfloor beta 2^{d-1} rfloor$ in the discrete hypercube $Q_d = {0,1}^d$ for any fixed $beta in [0,1]$ as $d to infty$, extending a result of Galvin for $beta in [1-1/sqrt{2},1]$. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard core model at any fixed fugacity $lambda>0$. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.
In this note, by the umbra calculus method, the Sun and Zagiers congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.
Let $mathcal{S}$ be a finite cyclic semigroup written additively. An element $e$ of $mathcal{S}$ is said to be idempotent if $e+e=e$. A sequence $T$ over $mathcal{S}$ is called {sl idempotent-sum free} provided that no idempotent of $mathcal{S}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum free sequence over $mathcal{S}$ of length over approximately a half of the size of $mathcal{S}$ is well-structured. This result generalizes the Savchev-Chen Structure Theorem for zero-sum free sequences over finite cyclic groups.
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