弗伦克尔猜想的一个证明
A Proof of Fraenkel's Conjecture
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中文总结 AI 辅助
本文证明了弗伦克尔猜想，即整数划分为至少3个不同模的贝蒂序列时密度符合特定二进制模式，通过密度界、傅里叶分析等方法完成，验证了猜想的正确性。
中文摘要 AI 辅助
弗伦克尔猜想断言：将整数划分为至少3个具有不同模的贝蒂序列时，其二进制密度为1,2,4,…,2^(m-1)，并按2^m-1归一化。我们通过一个与维度无关的中间命题证明该猜想：每一个此类划分都包含一个密度至少为1/3的分量。在将划分归约为本原公共周期数据后，傅里叶抵消会产生一个有限逆正弦方程组。我们证明当所有密度均低于1/3时，该方程组不可能存在。该证明结合了除子集中恒等式、一致解析估计以及三次精确有限验证，所有运算均采用整数或有理数算术完成。由密度界给出的分量平均间距至多为3；删除它后仍保持平衡，且每个剩余的周期平衡集再次为有理贝蒂集。归纳法确定剩余的二进制尺度，而两序列不相交准则迫使被删除的密度为下一个二进制尺度，由此得到断言的密度模式。
英文摘要
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3. After reducing the partition to primitive common-period data, Fourier cancellation produces a finite inverse-sine system. We prove that no such system can exist when every density is below 1/3. The proof combines a divisor-concentration identity with uniform analytic estimates and three exact finite verifications, all carried out with integer or rational arithmetic. The component supplied by the density bound has mean spacing at most three. Deleting it preserves balance, and every surviving periodic balanced set is again a rational Beatty set. Induction determines the surviving binary scales, while a two-sequence disjointness criterion forces the deleted density to be the next binary scale. This yields the asserted density pattern.An eventual Beatty representation of balanced binary indicators extends the classification to one-sided balanced sequences on at least three letters with distinct positive densities, proving the balanced-sequence conjecture of Altman, Gaujal, and Hordijk.
