Golomb标尺差集填充的几乎覆盖阈值

中文摘要 AI 辅助

对于固定整数t≥3，考虑t标记Golomb标尺的族，其正差集两两不交且包含于区间[1,U]。令P_t(U)为该族覆盖的最大整数个数。我们确定了渐近完全覆盖的阈值：P_t(U)=U−o(U)当且仅当3≤t≤5。t=3、4的情形由完美差族的已知存在谱得出。对于t=5，Wild的乘积构造（以Mathon记录的形式应用于阶数为121和161的完美族）给出了精确覆盖尺度的乘法半群；其对数上的初等密度引理则在每个足够大的U下方提供了一个尺度(1−o(1))U。对于逆命题，我们给出了一个自包含的单频傅里叶障碍。若x₀∈(π,3π/2)是tanx=x的第一个正解，且γ₀=−2sinx₀/x₀=0.4344672564…，则对于每个固定t≥6，有lim inf_{U→∞}(1−P_t(U)/U)≥((t−1)γ₀−2)/(2(t−2))。特别地，6标记标尺的强制间隙至少为2.1542035%。我们还证明了一个离散小差界，该界对每个t≥14产生更强的障碍，并在t→∞时强制间隙为1/2−1/√t−7/(8t)+O(t⁻³/²)。

英文摘要

For a fixed integer $t\geq 3$, consider families of $t$-mark Golomb rulers whose positive-difference sets are pairwise disjoint and contained in $[1,U]$. Let $P_t(U)$ be the largest number of integers covered by such a family. We determine the threshold for asymptotically complete coverage: \[ P_t(U)=U-o(U) \quad\Longleftrightarrow\quad 3\leq t\leq 5. \] The cases $t=3,4$ follow from the known existence spectra for perfect difference families. For $t=5$, Wild's product construction, in the form recorded by Mathon and applied to perfect families of orders $121$ and $161$, gives a multiplicative semigroup of exact-covering scales; an elementary density lemma on its logarithms then supplies a scale $(1-o(1))U$ below every sufficiently large $U$. For the converse, we give a self-contained one-frequency Fourier obstruction. If $x_0\in(π,3π/2)$ is the first positive solution of $\tan x=x$ and \[ γ_0=-\frac{2\sin x_0}{x_0}=0.4344672564\ldots, \] then, for every fixed $t\geq 6$, \[ \liminf_{U\to\infty}\left(1-\frac{P_t(U)}{U}\right) \geq \frac{(t-1)γ_0-2}{2(t-2)}. \] In particular, the forced gap for six-mark rulers is at least $2.1542035\%$. We also prove a discrete small-difference bound which yields a stronger obstruction for every $t\geq14$ and forces a gap of \[ \frac12-\frac1{\sqrt t}-\frac7{8t}+O(t^{-3/2}) \] as $t\to\infty$.

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9 pages, 2figures