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Register a new userSmall-Support Uncertainty Principles on $mathbb{Z}/p$ over Finite Fields
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We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.
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Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ over $F$ have rank $leq r$ ? This question is classical, and the answer ($q^{2r}$ when $rleqminleft{ p,qright} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},ldots,x_{k-1}$ for some $kleq rleqminleft{ p,qright} $, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},ldots,x_{p+q}$ such that the resulting Hankel matrix $left( x_{i+j}right) _{0leq ileq p, 0leq jleq q}$ has rank $leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.
Given $E subseteq mathbb{F}_q^d times mathbb{F}_q^d$, with the finite field $mathbb{F}_q$ of order $q$ and the integer $d ge 2$, we define the two-parameter distance set as $Delta_{d, d}(E)=left{left(|x_1-y_1|, |x_2-y_2|right) : (x_1,x_2), (y_1,y_2) in E right}$. Birklbauer and Iosevich (2017) proved that if $|E| gg q^{frac{3d+1}{2}}$, then $ |Delta_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| gg q^{frac{10}{3}}$, then $ |Delta_{2, 2}(E)| gg q^2$. In this paper, we present extensions and improvements of these results.
We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.
For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$ which surjects onto $mathrm{Sp}_{2g}(mathbb{Z}/ellmathbb{Z})$ must in fact equal all of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buiums $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weilands biring formalism, and Deligne--Illusie classes.
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