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Floor function and density !

Source: Romania TST 2015 Day 1 Problem 4

April 9, 2015

floor functionDensityKroneckeralgebranumber theory

Problem Statement

Let

k

be a positive integer congruent to

1

modulo

4

which is not a perfect square and let

a=\frac{1+\sqrt{k}}{2}

. Show that

\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}