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On identities concerning integer parts

Zichang Wang, Chengyang Wu, Bohan Yang

TL;DR

The paper investigates identities involving floor and fractional parts that characterize certain algebraic integers, extending the classic results tied to the golden ratio. It proves a main equivalence: for positive integers $l$ and $k$, the condition $\alpha^{l+k}-\alpha^{l}\in\mathbb{Z}\cap[1,2^{\frac{l}{k}})$ is equivalent to the floor identity $[[n\alpha^{l}]\alpha^{k}]+1=[n\alpha^{l+k}]$ for all nonzero $n$, and it extends this with three partial generalizations using a delta shift, a negative multiplier $-m$, and a polynomial-subsequence variant. The proofs rely on equidistribution on tori via Kronecker–Weyl and Weyl’s equidistribution criteria, analyzing orbits of sequences like $(\{n\alpha^{l}\},\{n\alpha^{l+k}\})$ and their intersections with carefully chosen regions. These results yield new characterizations of algebraic integers through floor identities and link circle-rotation dynamics with number-theoretic structure, while also posing open questions about higher-nested identities.

Abstract

In 2007 V. Zhuravlev discovered a family of identities concerning integer parts which are satisfied by the number $\frac{\sqrt{5}+1}{2}$. Some of these identities turned out to be characterization properties of the number $\frac{\sqrt{5}+1}{2}$. In this paper we generalize the simplest of these identities.

On identities concerning integer parts

TL;DR

The paper investigates identities involving floor and fractional parts that characterize certain algebraic integers, extending the classic results tied to the golden ratio. It proves a main equivalence: for positive integers and , the condition is equivalent to the floor identity for all nonzero , and it extends this with three partial generalizations using a delta shift, a negative multiplier , and a polynomial-subsequence variant. The proofs rely on equidistribution on tori via Kronecker–Weyl and Weyl’s equidistribution criteria, analyzing orbits of sequences like and their intersections with carefully chosen regions. These results yield new characterizations of algebraic integers through floor identities and link circle-rotation dynamics with number-theoretic structure, while also posing open questions about higher-nested identities.

Abstract

In 2007 V. Zhuravlev discovered a family of identities concerning integer parts which are satisfied by the number . Some of these identities turned out to be characterization properties of the number . In this paper we generalize the simplest of these identities.

Paper Structure

This paper contains 6 sections, 15 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Proof of Theorem \ref{['gen1']}
  4. Proof of Theorem \ref{['gen2']}
  5. Proof of Theorem \ref{['gen3']}
  6. Appendix: Weyl's equidistribution criterion

Key Result

Theorem 1.1

A real number $\alpha$ satisfies the identity for all $n\in\mathbb{Z}\setminus \{0\}$, if and only if $\alpha=\frac{\sqrt{5}+1}{2}$.

Figures (2)

  • Figure 1: The three possibilities to be excluded: the left image corresponds to the case when $\{1,\alpha^l,\alpha^{k+l}\}$ is $\mathbb{Q}$-linearly independent; the middle image corresponds to the case when the subtorus intersects the $y$-axis except the origin; and the right image corresponds to the case when the slope is not equal to $1$.
  • Figure 2: An illustration of the proof when $m=2$. The orbit should be equidistributed in the parallel segments, and spend the same amount of time in different regions. Here the green segments are as desired; but the blue segment with slope $1$ is cut by the border between regions at some rational point, and the first of the red segments with slope $3$ doesn't lie in the region.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • proof : Proof of the "only if" part of Theorem \ref{['main']}
  • ...and 18 more