Let $q=p^f$, and let $A\leq\mathbb{F}_q^\times$ be a multiplicative subgroup with $\mathbb{F}_p(A)=\mathbb{F}_q$. We prove that a proper subgroup $A$ is a generalized arithmetic progression (GAP) if and only if $|A| \in \{1, 2, 4\}$, and we determine when the full group $\mathbb{F}_q^\times$ is a GAP. For certain families of subgroups, we obtain the stronger conclusion that $A$ is additively irreducible. In particular, if $|A|>4$ and $p^e\equiv-1\pmod{|A|}$ for some $e\ge1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c \neq 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[\mathbb{F}_q^\times:A] \ge3$ and $|A| \ge 5$, which may be of independent interest.
Let $b, p\in {\mathbb Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1 \pmod{p^2}$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in {\mathb...
Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power...
Let $p$ be a prime and let $\mathcal{P}_p$ be the set of primitive elements of $\mathbb{F}_p$. Inspired by the work of Cohen, Oliveira e Silva and Trudgian on consecutive primitive elements, and by the work of Chang on arithmetic progressions in multiplicative subgroups of finite fields, in this paper, using estimates...
Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$....
Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd(a,b)=1$. For a finite set $A\subset\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A...
Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathb...
Maxim Kukol· 0 citations
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