发表机构
Academy of Mathematics and Systems Science Chinese Academy of Sciences; School of Mathematical Sciences Soochow University（中国科学院数学与系统科学研究院; 苏州大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明非负循环分圆倍式的支撑直径下界为(p-1)N/p，并刻画等式情形，证实Steinberger猜想，进而建立Coven-Meyerowitz直径界及其改进。
AI 中文摘要
设 \\(N\ge2\\)，并令 \\(p\\) 为其最小素因子。我们证明，每个具有非负实系数且能被 \\(\Phi_N\\) 整除的非零多项式，其支撑直径至少为 \\((p-1)N/p\\)。等式成立的情形恰好是 \\(p\\) 项几何和 \\(\sum_{j=0}^{p-1} X^{jN/p}\\) 的单项式平移的正标量倍数，从而证明了 Steinberger 的一个猜想。证明将循环分圆整除性转化为圆周上正测度的前 \\(p-1\\) 个傅里叶矩的消失，然后应用一个经典的极值三角多项式。作为推论，我们在 Coven--Meyerowitz 的平铺条件下建立了其直径界，并确定了其等式情形。更长的初始傅里叶系数消失区间产生更强的直径界，包括一个基于素幂因子集合的显式改进。极值三角多项式还给出了具有近最小支撑直径的测度和循环分圆倍式的定量集中估计。
英文摘要
Let \(N\ge2\) and let \(p\) be its least prime divisor. We prove that every nonzero polynomial with nonnegative real coefficients divisible by \(Φ_N\) has support diameter at least \((p-1)N/p\). Equality holds precisely for positive scalar multiples of monomial shifts of the \(p\)-term geometric sum \(\sum_{j=0}^{p-1} X^{jN/p}\), thereby proving a conjecture of Steinberger. The proof turns cyclotomic divisibility into the vanishing of the first \(p-1\) Fourier moments of a positive measure on the circle and then applies a classical extremal trigonometric polynomial. As a consequence, we establish the Coven--Meyerowitz diameter bound under their tiling conditions and determine its equality cases. Longer initial intervals of vanishing Fourier coefficients yield stronger diameter bounds, including an explicit refinement in terms of the prime-power divisor sets. The extremal trigonometric polynomial also yields a quantitative concentration estimate for measures and cyclotomic multiples with near-minimal support diameter.
Comments9 pages
