标题： 地板、天花板和它们之间的空间 显示英文标题

标题： Floor, ceiling and the space between

摘要： 受Gazeta Matematică中问题27327的启发，我们研究了依赖于实参数并通过实数的地板函数定义的函数的精确范围。 我们的发现表明，确定这些范围需要非平凡的工具，包括Kronecker逼近定理。 我们的结果强调了无理参数之间的深刻差异，并对比了单位以下和单位以上有理参数的行为。 我们还提出了与无理数和单位以上有理数相关的几个猜想，并通过使用Wolfram Mathematica进行的大规模计算得到了支持。 显示英文摘要

摘要： Inspired by a question posed in Problem 27327 from Gazeta Matematic\u{a}, we investigate the precise ranges of functions dependent on a real parameter and defined via the floor of a real number. Our findings reveal that determining these ranges necessitates non-trivial tools, including the Kronecker approximation theorem. Our results emphasize the deep distinctions among irrational parameters and contrast the behavior of sub-unitary and supra-unitary rational parameters. We also propose several conjectures related to irrational numbers and supra-unitary rational ones, supported by extensive computations using Wolfram Mathematica.