---

Let $m\geq1$ be an integer, put $d=2m+1$, and write $P$ for the permutation matrix that shifts the $d$ coordinates cyclically. Define

$$C_m=\left(I+\frac12P\right)^T.$$

Thus, with cyclic indices,

$$(C_mu)_0=u_0+\frac12u_{d-1},\qquad (C_mu)_i=u_i+\frac12u_{i-1}\quad(1\leq i<d).$$

Every column of $C_m$ has sum $3/2$.

Statement and proof

The only $z\in\mathbb Z^d$ with $\|C_mz\|_\infty<1$ is $z=0$. Multiplying each coordinate inequality by $2$ gives

$$|2z_i+z_{i-1}|<2$$

around the cycle, so [[additive_combinatorics/adamczewski_2026_erdos1/lemma_2_1|Lemma 2.1]] applies.

Source and dependencies

An explanation of the proof of Erdős Problem 1, preliminary exposition with no named author (erdosproblems.com, 2026), §2, equation (2) and Corollary 2.2, p. 2. The edition read is named on the source card.

Bears on. #1.