Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$\Omega(m)$ $\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \[ \frac{1}{K(K-1)} \sum_{\substack{1\le a,b\le K\\a\ne b}} \prod_{p\mid a^a-b^b}\left(1+\frac{\kappa}{p}\right) \le C_\kappa \] for some constant $C_\kappa>0$, which is valid for all $K \ge 2$ and any fixed $\kappa>0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.
Let $p \equiv 1$ (mod 4) be a prime number, and consider its representation as a sum of two squares,
$$p=a^2 + b^2,$$
for some integers $a$ and $b$. In this paper, we study the sequences $\{a_n\}$ and $\{b_n\}$ defined by the relation
$$p^n=a_n^2 + b_n^2,$$
arising from powers of the Gaussian integer $a+bi$.
In Sectio...
Arto Adili, Adrian Naço, Lorena Kelo· Far East Journal of Mathemat...· 0 citations
For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $\gamma_2(T)\le Cr$, and $\gamma_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]...
Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\...
We prove Hitchin's conjecture for $\mathfrak{sl}_n$ in every odd prime degree. More precisely, let $p=2k+1$ be prime and let $n\geq k+1$. Every nonzero primitive element $\alpha\in\Lambda^p(\mathfrak{sl}_n^*)^{\mathfrak{sl}_n}$ is nonzero on the top exterior power of the unique $p$-dimensional irreducible submodule of...
For a given number $N$, we consider the problem of computing two integers $1\leq r,f<N$ such that the set $$\mathcal{X}(N,r,f) = \{(a+b)-(f+\frac{Nr+1}{f}): ab=Nr\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying $l(Nr)<f \leq l(Nr+1)$, where...
We study the $\beta=2$ partition function $\int_{|h| \leq \log^{\theta}(q)/2}|L(1/2+ih,\chi)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $\theta \in (-1/2,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. When $\theta<0$, we use Harper's randomisation argument to introduce explic...
Christopher Atherfold· 0 citations
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