紧孤独赛跑者实例的单速度修改：有效界与r=2的完全分类
Tight instances of the Lonely Runner Conjecture: complete classification of one-entry modifications, a new infinite family, and the growth bound
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中文总结 AI 辅助
该研究改进孤独赛跑者猜想相关有限性界，完成r=2的单速度修改紧集分类，证明2r≤n-1时紧集的n≤6r，发现紧集速度可高于2n的反例。
中文摘要 AI 辅助
对于n-1个不同正整数构成的集合V，记LR(V)=max_t min_{v∈V} ||v t||，其中||x||为x到最近整数的距离；若LR(V)=1/n，则称V是紧的，这一值由孤独赛跑者猜想预测。Goddyn与Wong（《Integers》第6卷，2006年，#A38）对从基线[n-1]中替换一个速度r为其倍数mr得到的紧集进行了分类，并证明对于固定的r，仅存在有限个非倍数替换能构成紧集，他们指出这“部分解释”了为何两个零散紧集{1,3,4,7}和{1,3,4,5,9}（其中速度2被奇数替换）无类似物。我们将他们的有限性结果具体化并解决了他们指出的情况。设U(n,r)为从基线中删除速度r后留下的区域，我们根据算术量I(n,r)，在2r>n-1和2r≤n-1两种情况下精确计算U(n,r)每个连通分支的长度。由于插入的速度w必须保持1/n-接近整数的连通集长度不能超过2/(wn)，这给出了显式必要界w≤4rI/(2s-I)，其中s=n-r，因此：若([n-1]去掉{r})与{w}的并集是紧集且2r≤n-1，则n≤6r。这将Goddyn与Wong有限性定理中的隐含常数从12改进为6，使单速度修改的分类对每个n都成为有限计算。对r=2执行计算后，我们得到完全分类：([n-1]去掉{2})与{w}的并集（w>n-1）是紧集当且仅当(n,w)=(5,7)或(6,9)；相同方法完全解决了r=3的情况。我们还报告了精确有理算术下的穷举普查，并指出“紧集的所有速度都低于2n”这一自然猜测不成立，反例为n=32时的Goddyn-Wong集{1,...,29,31,90}。
英文摘要
For a set V of n-1 distinct positive integers write LR(V) = max_t min_{v in V} ||vt||, where ||x|| is the distance from x to the nearest integer; V is tight if LR(V) = 1/n, the value predicted by the Lonely Runner Conjecture. The baseline [n-1] = {1,...,n-1} is tight for every n, and Perarnau and Serra list the characterization of tight instances as Problem 1 of their survey, noting no further progress since Goddyn and Wong, who classified the multiple case and proved finiteness for each fixed deleted speed. We settle the one-entry case completely: ([n-1] minus {r}) union {w} with w > n-1 is tight if and only if either 2r > n-1, r divides w and the Goddyn-Wong gcd criterion holds, or (n,r,w) = (5,2,7) or (6,2,9). In particular no tight one-entry modification exists for 3 <= r <= (n-1)/2, and the only tight cases beyond the Goddyn-Wong multiples are the two sporadic sets of Wills, {1,3,4,7} and {1,3,4,5,9}. The proof is purely theoretical: the connected components of the uncovered region U(n,r) are determined exactly in both regimes 2r > n-1 and 2r <= n-1, yielding the effective bound w <= 4rI/(2s-I), with s = n-r and I the least integer of [s, n-1] coprime to r; a single inequality, proved via the Jacobsthal function, closes the mid-range without computation. Tight one-entry modifications also satisfy max V <= 0.60 n log n + 52 n unconditionally, with sharp leading constant 1/2 along n = p#+2 (p# the primorial), so no linear bound confines tight instances. We further isolate an explicit CRT doubling subfamily of the Goddyn-Wong multi-acceleration theorem and prove for two-speed replacements that no tight instance contains a removed speed q with 2 <= q <= n/10 when n >= max(40,10q), whatever the second removal and the inserted speeds. An exact census over all n <= 140, w <= 8n finds only two tight two-swaps, the Wills set {1,4,5,6,7,11,13} and the Goddyn-Wong doubling at n = 74.
发表机构
- Yangzhou University（扬州大学）
机构由 AI 辅助整理，请以论文原文为准。
