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$ \mathbb {Z}_m$ IN AVERAGE’
$\mathbb {Z}_m$ IN AVERAGE
Given a positive integer m, let 
$\mathbb {Z}_m$ be the set of residue classes mod m. For 
$A\subseteq \mathbb {Z}_m$ and 
$n\in \mathbb {Z}_m$, let 
$\sigma _A(n)$ be the number of solutions to the equation 
$n=x+y$ with 
$x,y\in A$. Let 
$\mathcal {H}_m$ be the set of subsets 
$A\subseteq \mathbb {Z}_m$ such that 
$\sigma _A(n)\geq 1$ for all 
$n\in \mathbb {Z}_m$. Let 
$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$
Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that 
$\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove 
$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$
and investigate parallel results on subtractive bases of 
$ \mathbb {Z}_m$.
Validating the theoretical work on Rayleigh–Taylor instability (RTI) through experiments with an exceptionally clean and well-characterized initial condition has been a long-standing challenge. Experiments were conducted to study the three-dimensional RTI of an SF
$_6$–air interface at moderate Atwood numbers. A novel soap film technique was developed to create a discontinuous gaseous interface with controllable initial conditions. Spectrum analysis revealed that the initial perturbation of the soap film interface is half the size of an entire single-mode perturbation. The correlation between the initial interface perturbation and Atwood numbers was determined. Due to the steep and highly curved feature of the initial soap film interface, the early-time evolution of RTI exhibits significant nonlinearity. In the quasi-steady regime, various potential flow models accurately predict the late-time bubble velocities by considering the channel width as the perturbation wavelength. Differently, the late-time spike velocities are described by these potential flow models using the wavelength of the entire single-mode perturbation. These findings indicate that the bubble evolution is influenced primarily by the spatial constraint imposed by walls, while the spike evolution is influenced mainly by the initial curvature of the spike tip. Consequently, a recent potential flow model was adopted to describe the time-varying amplitude growth induced by RTI. Furthermore, the self-similar growth factors for bubbles and spikes were determined from experiments and compared with existing studies, revealing that a large amplitude in the initial soap film interface promotes the spike development.
Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting 
$a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field 
$\mathbb {F}_q$ containing q elements with 
$2\nmid q$, we give the explicit value of the determinant 
$D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if 
$q=p$ is a prime greater than 
$3$, then 
$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$
where 
$(\frac {\cdot }{p})$ is the Legendre symbol and 
$h(-p)$ is the class number of 
$\mathbb {Q}(\sqrt {-p})$.
