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Kaneko, Suzuki and Tachiya, arXiv:2601.20743v1, Theorem 1, printed/PDF p. 3. Definitions are on pp. 2–3; the source record identifies the selected artifact and reading scope.

Definitions

Let $q>1$ be a Pisot or Salem number, of degree $d$ over $\mathbb Q$. The source includes rational integers $q\ge2$ among Pisot numbers. For an algebraic number $\alpha$, write $\mathrm h(\alpha)$ for the maximum absolute value of its conjugates over $\mathbb Q$. This is the source's boxed coefficient size, not logarithmic height.

For a sequence $c=(c(n))_{n\ge1}$, put

$$\mathcal N_c=\{n\ge1:c(n)\ne0\},\qquad \mathcal N_c(x)=\mathcal N_c\cap[1,x),\qquad S_c(x)=\sum_{1\le n<x}\mathrm h(c(n)).$$

For the distinguished real embedding and real $x>1$, $z\ge0$, set

$$R_c(q,x,z)= \sum_{\substack{n\in\mathbb Z\\1\le n<x}} \ \sum_{\substack{j\in\mathbb Z\\j\ge z}}|c(n+j)|q^{-j}.$$

The inner sum is infinite. For rational integers, $\mathrm h(c(n))$ equals $|c(n)|$.

Statement

Let $a,b$ be sequences of algebraic integers of $\mathbb Q(q)$ with $a(n)\ge0$ for all $n\ge1$ and $\mathcal N_a$ infinite. Suppose real sequences $x_j,y_j,z_j\ge1$ and a fixed $\eta\in(0,1]$ satisfy, as $j\to\infty$,

$$x_j\to\infty,\qquad S_a(x_j),S_b(x_j)=O(y_j),$$ $$\#\mathcal N_a(x_j),\#\mathcal N_b(x_j)=o(x_j/z_j),$$ $$R_a(q,\eta x_j,z_j),R_b(q,\eta x_j,z_j) =o(x_j/y_j^{d-1}).$$

If $\mathcal N_b$ is infinite, require constants $\Delta,L>1$ such that for every two consecutive elements $m<m_+$ of $\mathcal N_b$ and every real $\mu\ge L$,

$$m+\Delta\mu<m_+ \quad\Longrightarrow\quad \mathcal N_a\cap[m+\mu,m+\Delta\mu)\ne\varnothing. \tag{G}$$

Then the convergent series $\sum_{n\ge1}(a(n)+b(n))q^{-n}$ does not belong to $\mathbb Q(q)$. When $\mathcal N_b$ is finite, condition (G) is absent. The label (G) is this page's; the paper numbers the hypotheses (i)–(v), and (G) is its condition (v).

Proof pointer, standing and use

The proof on p. 11 combines Lemmas 1–3, pp. 6–11: rationality over the base field supplies a nonzero algebraic-integer tail with a lower norm bound; sparsity and the averaged estimate make many tails small; (G) supplies nonvanishing on enough of those indices.

The statement, its definitions and the full-tail convention were checked against the source. The proof route has been read, but no complete source-proof reconstruction or independent acceptance is recorded. The quantitative replacement of the tail hypothesis is Theorem 2.

Bears on. Problem 249 as a possible transformation criterion. Its dense numerator sequence does not meet the sparsity hypothesis.