Guancheng Pan | Searcharxiv

arXiv2026

Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture

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.

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arXiv2026

Contractibility of the complex of incompressible Seifert surfaces: the knot case of Kakimizu's problem

Let $K \subset S^3$ be a non-trivial knot and let $IS(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint representatives. Kakimizu proved that $IS(K)$ is connected; whether it is contractible was asked by Przytycki and Schultens, who also identified the obstruction, namely that the projection used in the connectedness proof is not known to be well defined on isotopy classes. We prove that $IS(K)$ is contractible, and likewise every truncation $IS_\ell(K)$ on the vertices of genus at most $\ell$. The argument factors through a purely combinatorial theorem: every non-empty connected flag exchange complex is contractible, where an exchange complex carries a complexity function subject to two axioms which require only that an exchanging vertex exist, never that one be selected. That is what circumvents the obstruction. We locate the combinatorial theorem precisely against the existing literature -- it is not implied by dismantlability, and, when all descending links are finite, it follows from Zaremsky's descending-link criterion by a reindexing (Section 7.4) -- and we record what remains genuinely open. Two extensions are proved: the truncations above, and the case of links satisfying a linking condition that forces every spanning surface to be connected.

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