Let P be a positive rational number. A function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R has the finite gaps property mod P if the following holds: for any positive irrational $$\alpha $$ α and positive integer M , when the values of $$f(m\alpha )$$ f ( m α ) , $$1\le m\le M$$ 1 ≤ m ≤ M , are inserted mod P into the interval [0, P ) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant $$k_{f}$$ k f which depends only on f . In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non-differentiable points has the finite gaps property mod P . We also show that if f is the distance to the nearest integer function, then it has the finite gaps property mod 1 with $$k_f\le 6$$ k f ≤ 6 .

中文翻译：

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3d 距离定理的推广

令 P 为正有理数。函数 $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R 具有有限间隙属性 mod P，如果满足以下条件：对于任何正无理数 $$\alpha $$ α 和正整数 M ，当 $$f(m\alpha )$$ f ( m α ) , $$1\le m\le M$$ 1 ≤ m ≤ M 的值被 mod P 插入到区间 [0, P ) 并按递增顺序排列，连续项之间的不同间隙的数量受常数 $$k_{f}$$ kf 的限制，该常数仅取决于 f 。在本笔记中，我们证明了 Chung 和 Graham 的 3d 距离定理的推广。因此，我们证明了具有有理斜率且只有有限多个不可微点的分段线性映射具有有限间隙属性 mod P 。我们还表明，如果 f 是到最近的整数函数的距离，