Let $H \leq \mathbb{F}_p^*$ be a proper multiplicative subgroup, and suppose that $H = A+B$ for some $A,B \subseteq \mathbb{F}_p$. We prove that either one of the summands is a singleton, or $|A|=|B|=2$ and $|H|=4$. In particular, no proper multiplicative subgroup of $\mathbb{F}_p^*$ can be written as $A+B$ with $|A|,|B|>2$. Our proof builds on the Hanson-Petridis polynomial method and Kalmynin's subsequent resolution of S\'ark\"ozy's conjecture for quadratic residues. Using Kalmynin's $|A|=|B|$ theorem as a structural input, we develop uniform combinatorial and arithmetic arguments which apply to multiplicative subgroups of arbitrary index.
Let $p$ be an odd prime and let $A,B\subseteq\mathbb{F}_p$ be nonempty and distinct, with $|A|\ge |B|$. We give a complete classification of the pairs for which the mixed restricted sumset $A\mathbin{\widehat+} B=\{x+y:x\in A,\ y\in B,\ x\ne y\}$ has size $\min\{p,|A|+|B|-2\}$. The main new contribution concerns the in...
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 $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^...
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...
Lenny Jones· 0 citations
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