Let $u=(u_n)_{n\ge0}$ be a binary sequence, and define \[ X_0=1,\qquad X_{n+1}=2X_n+u_n \qquad(n\ge0). \] Let $A_u$ be the set of all nonempty finite subset sums of the sequence $(X_n)$, and let $L_u(N)$ denote the maximum length of an arithmetic progression contained in $A_u\cap[1,N]$. We prove that there are absolute...
Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb{F}_p^{2m}$ with Witt index $m-1$. For a nonempty set $E\subseteq\mathbb{F}_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|\geq p^m$, $$ |\Delta_Q(E)|\gg \frac{p}{\log\bigl(2+p^{m...
Thang Pham, Chun-Yen Shen, Dung The Tran et al.· 0 citations
Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\inf...
Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \ri...
Thang Pham, A. Pinamonti, Dung The Tran et al.· 0 citations
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