../

---

Source. Y. Yu and K. Chen, Erdős Problem 354(i): Strong Completeness of Two Dyadic Floor Sequences, manuscript of 13 September 2026, Lemma 2.3 with its display (2.3), physical p. 4, in the seventeen-page PDF held by its library source card, Yu and Chen (2026).

Standing. This is an author-recorded reconstruction. It is not an independent review, changes no status and assigns no tier.

Definitions

$\operatorname{span}$ and $\operatorname{gap}$ of a finite integer set with at least two elements are as on the mesh lemma page, and $h$ of a nonempty subset of $\mathbb Z/m\mathbb Z$ (the longest run of missing residues) is as on the erosion lemma page. For $W\subseteq\mathbb Z$, $W\bmod m$ is its image in $\mathbb Z/m\mathbb Z$.

Statement

Lemma 2.3. If $\operatorname{span}(W)\ge m\ge1$ and $\operatorname{gap}(W)\le k$, then

$$h(W\bmod m)\le k-1.$$

Proof

Translating $W$ by an integer rotates $W\bmod m$ by a fixed residue and changes neither $h$, the span nor the gap, so assume $\min W=0$. The set $W$ has at least two elements, so $k\ge1$.

If $m=1$, then $\mathbb Z/1\mathbb Z$ has one residue, $W\bmod 1$ is full, and $h=0\le k-1$.

Let $m\ge2$. Let $w_-$ be the largest element of $W$ in $[0,m)$; it exists because $0\in W$. Let $w_+$ be the smallest element of $W$ with $w_+\ge m$; it exists because $\max W=\operatorname{span}(W)\ge m$. No element of $W$ lies strictly between $w_-$ and $w_+$, so they are consecutive in $W$ and $w_+-w_-\le k$, whence

$$m-w_-\le w_+-w_-\le k.$$

Now consider the residues of the points of $W\cap[0,m)$; these integers are their own residues, and they include $0$ and $w_-$. Any two consecutive ones differ by at most $k$, so between them at most $k-1$ residues are missing. The remaining residues are $w_-+1,\ldots,m-1$, a run of length $m-1-w_-\le k-1$, and it is followed cyclically by the residue $0$, which is present. Hence every maximal run of residues missing from $W\cap[0,m)\bmod m$ has length at most $k-1$. The residues of the other points of $W$ can only shorten missing runs, so $h(W\bmod m)\le k-1$.