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In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyv{a}ri and Hennecart.
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Hegyvari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $Bcdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.
We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_infty^m(q^t;q^t)_infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)leq 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.
Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) in G^3$ satisfying $b-a = c-b eq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Omega_2(G)$ has the same cardinality as $G$, where $Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = mathbb{Z}/nmathbb{Z}$ for all $n eq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(mathbb{Z}/pmathbb{Z})^k$ if and only if $p$ is odd and $(p,k) otin {(3,1),(5,1), (7,1)}$.
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$ and all column sums in every rectangle equal to a constant $delta in Gamma$. In this paper we prove that for ${a,b} eq{2^{alpha},2k+1}$ where $alpha$ and $k$ are some natural numbers, a $Gamma$-magic rectangle set MRS$_{Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Gamma|$ is odd or $Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Gamma$-magic rectangle MRS$_{Gamma}(a, b)$=MRS$_{Gamma}(a, b;1)$.
Let $(R, mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/mathfrak{m} = mathbb{F}_{q}$. Given a monic polynomial $P(t) in R[t]$ whose reduction modulo $mathfrak{m}$ gives an irreducible polynomial $bar{P}(t) in mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $mathrm{coker}(P(A))$, where $A in mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $mathrm{Mat}_{n}(R)$ of $n times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $bar{P}$-part of a random matrix $bar{A} in mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime $p$, any finite abelian $p$-group (i.e., $mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $mathbb{Q}$ with a probability inversely proportional to $|mathrm{Aut}_{mathbb{Z}}(H)|$. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between $mathrm{Mat}_{n}(R)$ and $mathrm{Mat}_{n}(mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $mathrm{Mat}_{n}(R)$ into problems about a random matrix in $mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution.
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