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Retain the odd dimension $d$ and cyclic matrix $C_m$ from [[additive_combinatorics/adamczewski_2026_erdos1/corollary_2_2|Corollary 2.2]].

Statement

For a real $u_0$ and integers $u_1,\ldots,u_{d-1}$, the bound $\|C_mu\|_\infty<1$ forces

$$|u_0+u_1+\cdots+u_{d-1}|<1.$$

Proof

Abbreviate the integer coordinates as $z_i=u_i$ ($i\geq1$). For an index $j$ at which $|u_j|$ is largest, the inequality $|2u_j+u_{j-1}|<2$ yields

$$2|u_j|\leq|2u_j+u_{j-1}|+|u_{j-1}|<2+|u_j|,$$

so every coordinate lies in $(-2,2)$ and every $z_i$ lies in $\{-1,0,1\}$.

The real coordinate also satisfies $|u_0|<1$. Suppose $u_0\geq1$; then $|2u_0+z_{d-1}|<2$ forces $z_{d-1}=-1$, while $|2z_1+u_0|<2$ forces $z_1\in\{-1,0\}$. Define the integer vector

$$w_0=1,\qquad w_i=z_i\quad(1\leq i<d).$$

At index $0$, $|2w_0+w_{d-1}|=1$; at index $1$, $|2w_1+w_0|=1$; and at all remaining indices the original inequalities apply. This contradicts [[additive_combinatorics/adamczewski_2026_erdos1/lemma_2_1|Lemma 2.1]], since $w_0=1$. So $u_0<1$, and since $-u$ satisfies the same hypotheses, also $u_0>-1$.

For each $1\leq i<d-1$, checking the nine possible pairs under

$$|2z_{i+1}+z_i|<2,\qquad z_i,z_{i+1}\in\{-1,0,1\},$$

shows that $z_{i+1}=0$ or $z_{i+1}=-z_i$. Once a zero occurs, every later term is zero. Thus the nonzero terms of the tail $z_1,\ldots,z_{d-1}$ form an alternating initial segment $z_1,\ldots,z_p$, whose consecutive pairs cancel, so the tail sums to $0$ when $p$ is even and to $z_1$ when $p$ is odd.

If the tail sums to $0$, the total is $u_0$, of absolute value below $1$. In the other case the tail sum is $z_1$. If $z_1=1$, then $|2z_1+u_0|<2$ gives $u_0<0$, which together with $|u_0|<1$ gives $-1<u_0<0$ and hence $0<u_0+z_1<1$. If $z_1=-1$, it gives $u_0>0$, hence $0<u_0<1$ and $-1<u_0+z_1<0$. The case $z_1=0$ belongs to the first alternative. Therefore the total coordinate sum always has absolute value less than $1$.

Source and dependencies

An explanation of the proof of Erdős Problem 1, preliminary exposition with no named author (erdosproblems.com, 2026), §2, Lemma 2.3, pp. 2–3. The edition read is named on the source card. The final two sign cases make explicit the source's last one-line estimate.

Bears on. #1.