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Statement

Setting (pp. 1723, 1739). A subset $\mathcal A$ of a finite abelian group $G$ is foncièrement générateur when $i\mathcal A=G$ for some integer $i$. For integers $h\ge1$ and $c\ge2$, $K_2(h,c)$ is the largest integer $k$ for which there are an integer $g$ and a foncièrement générateur subset $\mathcal A$ of $\mathbb Z/g\mathbb Z$ with $c$ elements such that

(i) $\mathcal A\cup2\mathcal A\cup\cdots\cup h\mathcal A=\mathbb Z/g\mathbb Z$, and

(ii) $(k-1)\mathcal A\ne\mathbb Z/g\mathbb Z$;

such a $k$ satisfies $k\mathcal A=\mathbb Z/g\mathbb Z$. Then $K(h)=\max_{c\ge2}K_2(h,c)$.

Théorème 20 (p. 1739). For every positive integer $h$, $K_2(h,2)\ge\left[\frac{h(h+4)}{3}\right]$.

Lemme 26 (p. 1756). For every positive integer $h$, $X(h)\ge K(h)$, with $X$ as defined on the Théorème 1 page.

The construction (p. 1756): for a foncièrement générateur $\mathcal A\subset\mathbb Z/g\mathbb Z$ meeting (i) with $(K(h)-1)\mathcal A\ne\mathbb Z/g\mathbb Z$, the set $\mathcal B=\{0\}\cup(\mathcal A+g\mathbb N)$, the residues of $\mathcal A$ lifted to integers, is a basis of order at most $h$, and $\mathcal B\setminus\{0\}$ is a basis of order exactly $K(h)$. The example (1.5) on p. 1719, $\{0\}\cup(\{2,5\}+11\mathbb N)$, which gives $X(4)\ge10$, has this shape.

Théorème 20 is proved (pp. 1739--1741) by exhibiting, for $g=[h(h+4)/3]+1$, an integer $x_0$ with $x_0-1$ prime to $g$ such that $\{1,x_0\}$ meets (i). Conjecture 21 (p. 1741) states $K_2(h,2)=[h(h+4)/3]$ for every positive $h$; Table 1 (p. 1742) lists exhaustively computed values of $K_2(h,c)$ for small $h$ and $c$, none exceeding $K_2(h,2)$.

Read depth

Claims checked: the definitions, Théorème 20 and Lemme 26 were read clause by clause on the page images of pp. 1739 and 1756, and the proof of Lemme 26 was followed. The case computations in the proof of Théorème 20 were not checked. Nothing here is independently reviewed.

Dependencies

None in the corpus. Together they give the lower bound of Théorème 1 (p. 1756).

Source. Alain Plagne, À propos de la fonction X d'Erdős et Graham, Annales de l'Institut Fourier 54 (2004), no. 6, 1717--1767; the edition read is named on the source card.

Bears on

• Problem 336: the construction produces bases $\mathcal B\setminus\{0\}$ of exact order $K(h)\ge[h(h+4)/3]$ such that $\mathcal B$ has exact order at most $h$; the paper states these as lower bounds for $X(h)$ and states no relation to the problem's $h(r)$.