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Register a new userOn a theorem of Hegyv{a}ri and Hennecart
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In this paper, we study growth rate of product of sets in the Heisenberg group over finite fields and the complex numbers. More precisely, we will give improvements and extensions of recent results due to Hegyv{a}ri and Hennecart (2018).
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In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 times infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Eulers Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Eulers Pentagonal Number Theorem along with an infinite family of generalizations.
We obtain a unification of two refinements of Eulers partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodts insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the sets $A$ and $B$. Our theorem considers a general linear map $L: mathbb{F}_p^n to mathbb{F}_p^m$, and subsets $A_1, ldots, A_n subseteq mathbb{F}_p$, and gives a lower bound on the size of $L(A_1 times A_2 times ldots times A_n)$ in terms of the sizes of the sets $A_1, ldots, A_n$. Our proof uses Alons Combinatorial Nullstellensatz and a variation of the polynomial method.
We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.
Let G be the Tate module of a p-divisble group H over a perfect field k of characteristic p. A theorem of Scholze-Weinstein describes G (and therefore H itself) in terms of the Dieudonne module of H; more precisely, it describes G(C) for good semiperfect k-algebras C (which is enough to reconstruct G). In these notes we give a self-contained proof of this theorem and explain the relation with the classical descriptions of the Dieudonne functor from Dieudonne modules to p-divisible groups.
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