We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$, and obtain the bound $[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leq\varphi(N)$ for every odd $k$. For the first coefficient, exact computation for $3\leq N\leq 30$ gives the full degree $\varphi(N)$ in every case, motivating the Chandra--Jain conjecture that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=\varphi(N)$ for all $N\geq 3$. We then reduce the higher modes to the coprime case and discuss the remaining degree question. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers. The Chandra--Jain conjecture has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814).
We introduce a family of discrete hyperbolic secant distributions on $\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gau\ss'constant $G=\varpi/\pi$, where $\varpi=\Gamma^2(1/4)/(2\sqrt{2\pi})$ is the lemniscate constant. Using the elliptic lambda-...
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
For integers $h,k\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\mathcal{R}(h,k)=\{|hA|:A\subset\mathbb{Z}, |A|=k\}$. Previously, the fixed-cardinality exponent law was known only for $k\leq 3$; every fixed $k\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality ex...
We study the central Hardy--Littlewood maximal operator $M_c$ in $\mathbb{R}^n$ for all $n\geq1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ such that $\|f-g\|_1>0$ and $M_cf=M_cg$. Along the way, we o...
We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an ass...
J. Jorgenson, Anders Karlsson, Lejla Smajlović· 0 citations
We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397--1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of...
