.png){kind=icon}
.png){kind=icon}
Return
Difference between two floor functions
.png){kind=icon}
PRE
AI
DOI：10.1007/s00605-026-02173-7.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
In this paper, we confirm several conjectures of Benyi and Curgus related to the description of the set Range(alpha) attained by the differences of two floor functions [alpha(2)n] and [alpha [alpha n]], where alpha is a fixed positive number and n runs through all positive integers. In particular, we show that for each irrational number alpha > 0 the set Range(alpha) is the largest possible and takes all integral values between 0 and [alpha ]+1 except possibly for some quadratic algebraic numbers alpha > 1 when Range(alpha) can be smaller and take only integral values between 1 and [alpha]. Moreover, for every quadratic algebraic number alpha > 0, we explicitly determine the set Range(alpha) as well. In both cases, t = 0 and t = [alpha] + 1, we give explicit conditions describing whether t is or is not in Range(alpha). Those conditions are given in terms of the coefficients u, v of the minimal polynomial x(2)-ux-v E Q[x] of alpha over Q. All these results allow us to determine the set Range(alpha) explicitly for every irrational number alpha > 0. We also describe all the cases when Range(alpha) is a singleton set.
Keywords:
Kronecker's approximation theorem
Density
Distribution modulo 1
Quadratic irrationality