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Register a new userAn improved spectral large sieve inequality for $SL_3(mathbb{Z})$
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Matthew P. Young
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We prove an improved spectral large sieve inequality for the family of $SL_3(mathbb{Z})$ Hecke-Maass cusp forms. The method of proof uses duality and its structure reveals unexpected connections to Heath-Browns large sieve for cubic characters.
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Let $F(boldsymbol{x})$ be a diagonal integer-coefficient cubic form in $min{4,5,6}$ variables. Excluding rational lines if $m=4$, we bound the number of integral solutions $boldsymbol{x}in[-X,X]^m$ to $F(boldsymbol{x})=0$ by $O_{F,epsilon}(X^{3m/4 - 3/2 + epsilon})$, conditionally on an optimal large sieve inequality (in a specific range of parameters) for approximate Hasse-Weil $L$-functions of smooth hyperplane sections $F(boldsymbol{x})=boldsymbol{c}cdotboldsymbol{x}=0$ as $boldsymbol{c}inmathbb{Z}^m$ varies in natural boxes. When $m$ is even, these results were previously established conditionally under Hooleys Hypothesis HW. Our $ell^2$ large sieve approach requires that certain bad factors be roughly $1$ on average in $ell^2$, while the $ell^infty$ Hypothesis HW approach only required the bound in $ell^1$. Furthermore, the large sieve only accepts uniform vectors; yet our initially given vectors are only approximately uniform over $boldsymbol{c}$, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply. In an appendix, we suggest a framework for non-diagonal cubics, up to Hessian issues.
In this paper we obtain a decoupling feature of the random interlacements process $mathcal{I}^u subset mathbb{Z}^d$, at level $u$, $dgeq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $textsf{F}$ and its translated $textsf{F}+x$, can be coupled with high probability of success, when $|x|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $textsf{F}$ and $textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].
In this paper we describe the local limits under conjugation of all closed connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology.
Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{frac{q^2 -1}{3}+1} +x$ over $mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{frac{q^2 -1}{d}+1} -bx$ over $mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form [ f(x)=(ax^{q} +bx +c)^r phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~text{over $mathbb{F}_{q^2}$}, ] where $a,b,c,u,v in mathbb{F}_{q^2}$, $r in mathbb{Z}^{+}$, $phi(x)in mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.
The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.
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