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Use $B,R,L_B,E_B$, and $\Phi_t$ from [[additive_combinatorics/adamczewski_2026_erdos1/lattice_reduction|the lattice reduction]] and [[additive_combinatorics/adamczewski_2026_erdos1/lemma_5_1|Lemma 5.1]]. Thus every nonzero $z\in\mathbb Z^n$ satisfies $\|L_B(z)\|_\infty\geq R$.

Statement

Let $t\geq E_B$ be an integer and $q_0$ a real number with

$$q_0\leq(t-E_B)R.$$

Then every nonzero $z\in\mathbb Z^n$ satisfies

$$\|\Phi_t(z)\|_\infty\geq q_0.$$

Proof

If $q_0\leq0$, the conclusion follows from nonnegativity of the norm. Suppose therefore that $q_0>0$. Let $z\ne0$ and put $M=\|L_B(z)\|_\infty\geq R$. Fix an index $j$ with $|L_B(z)_j|=M$. From $\Phi_t=tL_B+E$ and Lemma 5.1,

$$tM =|tL_B(z)_j| \leq|\Phi_t(z)_j|+|E(z)_j| \leq|\Phi_t(z)_j|+E_BM.$$

Were $\|\Phi_t(z)\|_\infty<q_0$, it would follow that

$$(t-E_B)M<q_0.$$

But $M\geq R$ and $t-E_B\geq0$, so the left side is at least $(t-E_B)R\geq q_0$, a contradiction.

Source and dependencies

An explanation of the proof of Erdős Problem 1, preliminary exposition with no named author (erdosproblems.com, 2026), §5, Proposition 5.2, p. 7. The edition read is named on the source card. Dependencies are Lemmas 4.1 and 5.1 and the triangle inequality.

Bears on. #1.