For an integer $k \ge 0$, let $f_k(x) = 1 + e(x) + e((k+2)x)$ and $g_k(x) = 1 + e(x) - e((k+2)x)$ on $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, where $e(x) = e^{2\pi i x}$. Mockenhaupt conjectured that $\|g_k\|_{L^p(\mathbb{T})}>\|f_k\|_{L^p(\mathbb{T})}$ whenever $2k<p<2k+2$. The conjecture was previously known for $k \le 5$. We give a single analytic proof valid for every $k \ge 4$; in particular, this settles all previously open cases $k \ge 6$ and establishes the conjecture for every $k \ge 0$. The proof reduces the norm comparison to resonant Fourier coefficients on the two-torus and represents these coefficients, after analytic continuation, by triple-Bessel integrals. Neumann's product formula and the Weber-Schafheitlin formula yield a quantitative positive lower bound for the leading mode, while the remaining odd modes are controlled by a uniform tail estimate. The leading mode is then shown to dominate the tail for every $k \ge 4$. AI Usage. The mathematical argument of this paper was produced by the auto-research system Apex Math, an AI system built by Apex Intelligence. See Appendix A for the complete AI usage statement.
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resol...
For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the q...
For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result...
Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants govern...
For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $\theta(s;z)=\sum_{m,n\in\mathbb Z}e^{-\pi s\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-\pi \big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2\pi(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respec...
For an integer solution of $y^2=x^3+k$ with $x,y\ge 1$ and $k\ne 0$, the integers $x^3$, $|k|$ and $y^2$ form a triple in which one term is the sum of the other two. When $\gcd(x,y)=1$ this triple is coprime, and the $abc$ conjecture predicts that its quality is asymptotically at most $1$. Following M\"uller, Taktikos...
R. Laniewski, K. Muller· 0 citations
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