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Source. Pipeline-math, Erdős problem 477, commit 99d916ff32a90e77c98eb004537ccda409262346 (29 June 2026), Proposition 1.8, printed/PDF p. 6 of the manuscript.

Statement

Let $B=\{m^{13}:m\in\mathbb Z\}$ and $D=B-B$. For every finite $C\subseteq\mathbb Z\setminus B$, there is a $b\in B$ such that

$$(C-b)\cap D=\varnothing.$$

This is exactly the hypothesis of Lemma 1.7 for this particular $B$.

Proof

Fix a finite set $C\subseteq\mathbb Z\setminus B$. If $C$ is empty, take $b=0\in B$. Suppose now that $C$ is nonempty. With $b=t^{13}$, the element $c\in C$ violates $(C-b)\cap D=\varnothing$ exactly when $c-t^{13}\in D$. The difference set is symmetric: if $u-v\in D$ with $u,v\in B$, then its negative is $v-u\in D$. Thus the same condition is $t^{13}-c\in D$.

For each fixed $c\in C$ and integer $T\ge1$, the integers $t$ with $|t|\le T$ at which $c$ violates that condition form the set

$$S_c(T)=\{t\in\mathbb Z:|t|\le T,\ t^{13}-c\in D\}.$$

By Proposition 1.6, there is a constant $K_c$ independent of $T$ such that $|S_c(T)|\le K_cT^{5/6}$. Since $C$ is fixed and finite,

$$\left|\bigcup_{c\in C}S_c(T)\right| \le\sum_{c\in C}|S_c(T)| \le K_CT^{5/6},\qquad K_C=\sum_{c\in C}K_c<\infty.$$

There are $2T+1$ integers with $|t|\le T$ because we take $T$ integral. As $T\to\infty$ along the integers, $K_CT^{5/6}/(2T+1)\to0$. Hence for some sufficiently large $T$ at least one integer $t$ in the interval avoids every $S_c(T)$. With $b=t^{13}$, no $c-b$ belongs to $D$, as required.

The choice of $T$ and $b$ may depend on the whole finite set $C$. The proof uses a finite sum of fixed-shift estimates; it asserts neither an infinite-union estimate nor a bound uniform over all shifts.

Dependencies and current verification

This complete reconstruction consumes Proposition 1.6 and its stated external premises. The [[diophantine_problems/pipeline_math_2026_tiling_complement/evidence/verify/compilation_review|independent compilation review]] found no material defect in the exact frozen statement, essential deductions and their composition. The source's statement and proof on p. 6 were read in text and rendered images. Taking integer $T$ supplies the source's exact count $2T+1$; for a real parameter the count would be $2\lfloor T\rfloor+1$. Attack selection was partly pre-directed; the derivations were independently performed. The six-result review is relative to the Corvaja-Zannier-recalled unit bounds and Heath-Brown's journal Theorem 2, with the recorded nonconstant-family qualification. The external proofs were not independently reviewed; no formal verification is claimed. See the [[diophantine_problems/pipeline_math_2026_tiling_complement/_index|source digest]].

Bears on. The proposition supplies the premise used by Theorem 1.1 to answer Problem 477.