We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We show that for every pair $(a,b)$ of positive integers, all the binary balanced words with $a$ zeroes and $b$ ones are good approximations of the Euclidean segment from $(0,0)$ to $(a,b)$, in the sense that they encode paths that are contained within the region of the grid delimited by the lower and the upper Christoffel words of slope $b/a$. We then give a closed formula for counting the exact number of balanced words with $a$ zeroes and $b$ ones. We also study minimal non-balanced words and prefixes of Christoffel words.

翻译：我们展示了关于Christoffel词和二元平衡词的组合结果，这些结果源于其作为数字线段近似几何解释的推动。我们证明，对于每一对正整数$(a,b)$，所有包含$a$个零和$b$个一的二元平衡词，都是对从$(0,0)$到$(a,b)$欧几里得线段的良好近似，其意义在于它们编码的路径被包含在斜率为$b/a$的下Christoffel词和上Christoffel词所界定的网格区域内。随后，我们给出一个闭合公式，用于精确计算包含$a$个零和$b$个一的平衡词的数量。我们还研究了最小非平衡词以及Christoffel词的前缀。