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Statement

For a set $\mathcal A$ of integers, $h^\wedge\mathcal A$ is the set of integers representable as a sum of $h$ distinct elements of $\mathcal A$, and $\mathcal A$ is admissible when $s^\wedge\mathcal A\cap t^\wedge\mathcal A=\emptyset$ for all $s\ne t$ (printed p. 33).

Theorem 2 (printed p. 34). "Let $\mathcal A$ be an admissible set included in $[1,N]$, such that $\operatorname{card}\mathcal A>1.96\sqrt N$. If $N$ is large enough, there exists $\mathcal C\subset\mathcal A$ having the following properties:

• (i) $\operatorname{card}\mathcal C\le10^5N^{5/12}$,
• (ii) for some $t$, the set $t^\wedge\mathcal C$ contains an arithmetic progression with at least $3N^{5/6}$ terms, and difference $d$, say,
• (iii) $\mathcal A\setminus\mathcal C$ is included in an arithmetic progression with difference $d$, and containing at most $N^{7/12}$ terms."

Remark (p. 34, quoted). "It will be clear from the proof that a similar result may be obtained when 1.96 is replaced by any number larger than $4\sqrt{2/3}=1.8856\ldots$." Filing observation (PDF p. 2, page image at 300 dpi: the root sign covers $2/3$): the printed expression does not equal the printed value, since $4\sqrt{2/3}=3.2659\ldots$; the constant intended is $4\sqrt2/3=1.8856\ldots$, which matches the printed value. The paper introduces the theorem as "a first step" toward the structure of large admissible sets, "however far from being stated in its strongest shape", and says that Theorem 1 "is an easy consequence of it".

Theorem 3 (p. 34, quoted), the inverse result behind it, "a consequence of the structural result of the second author": "Let $\lambda<6$ and $\mathcal B$ be a finite set of integers such that $\operatorname{card}(4^\wedge\mathcal B)\le\lambda\operatorname{card}\mathcal B$. There exist real numbers $C_1(\lambda)$ and $C_2(\lambda)$ such that $\lfloor(C_1\operatorname{card}\mathcal B)^\wedge\mathcal B\rfloor$ contains an arithmetic progression with at least $C_2(\lambda)(\operatorname{card}\mathcal B)^2$ terms." Only the special case $\lambda=5.8$ (Proposition 4, p. 38) is proved, "which is enough for our purpose".

The 1999 quotation. The sequel's Theorem 1 page quotes this theorem as its Theorem 2, with the progressions in (ii) and (iii) described as arithmetic progressions "modulo $q$" in place of "with difference $d$"; the constants $1.96$, $10^5N^{5/12}$, $3N^{5/6}$ and $N^{7/12}$ are the same. A filing observation, not a review verdict.

Source. J-M. Deshouillers and G. A. Freiman, On an additive problem of Erdős and Straus, 1, Israel J. Math. 92 (1995), 33--43, doi:10.1007/BF02762069; Theorem 2, its remark and Theorem 3 on printed p. 34 (PDF p. 2) of the publisher's PDF, read on the page image; the proof on printed pp. 35--41 (PDF pp. 3--9). The artifact is identified in the source digest.

Read depth. Claims checked: the statement, the remark and Theorem 3 were read clause by clause on the page image, and the standing assumption $1.96\sqrt N\le\operatorname{card}\mathcal A\le2.31\sqrt N$ (pp. 34--35) with it. Proposition 1 and its proof (p. 35) were read on the page image; Sections 2--5 (pp. 35--41), the proof proper, were read in the OCR text layer for structure only (p. 41 on the page image), and none of the numerical constants was checked. Nothing here is independently reviewed.

Proof pointer

Sections 1--5 (pp. 35--41), under the standing assumption $1.96\sqrt N\le\operatorname{card}\mathcal A\le2.31\sqrt N$ (the upper bound from Straus). Proposition 1 (p. 35): some $s\in[|\mathcal A|/10,3|\mathcal A|/4]$ has $|s^\wedge\mathcal A|<1.44s(|\mathcal A|-s)$, since the sets $s^\wedge\mathcal A$ over that range are disjoint inside $[1,0.75|\mathcal A|N]$. Proposition 2 (pp. 35--36): for $1\le L\le|\mathcal A|/2000$ there is $\mathcal B\subset\mathcal A$ with $|\mathcal B|=L$ and $|4^\wedge\mathcal B|<5.8L$, found in a block $\mathcal C_l$ of $s+4$ consecutive elements of $\mathcal A$ with small $|s^\wedge\mathcal C_l|=|4^\wedge\mathcal C_l|$. Section 3 collects Freiman's inverse theorem in its easiest case (Proposition 3.1), a lemma on $h\mathcal S$ for a set inside a progression of length at most $1.94|\mathcal S|$ (Proposition 3.2) and $|2\mathcal B|\le3|\mathcal B|+|4^\wedge\mathcal B|$ (Proposition 3.3). Proposition 4 (p. 38): for $L$ large and $|4^\wedge\mathcal B|\le5.8L$, the set $2\lfloor L10^{-6}\rfloor^\wedge\mathcal B$ contains at least $10^{-8}|\mathcal B|^2$ terms of an arithmetic progression, via the set $\mathcal S$ of elements of $2\mathcal B$ with many representations, which Proposition 3.1 puts in a short progression. Section 5 (pp. 39--41) takes $L:=2\lfloor10^4N^{5/12}\rfloor$ and $t:=2\lfloor10^{-6}L\rfloor$, gets a progression of difference $\delta$ and at least $3N^{5/6}$ terms in $t^\wedge\mathcal B$, shows that $\mathcal A\setminus\mathcal B$ meets fewer than $\lfloor N^{1/6}\rfloor$ residue classes modulo $\delta$ and that the differences between one of its "rich" classes and each of the others have order less than $\lfloor N^{1/6}\rfloor$ in $\mathbb Z/\delta\mathbb Z$ and generate a subgroup $G$ (p. 40), sets $d:=\delta/|G|$, and finally moves the $\lfloor3N^{5/12}\rfloor$ smallest and largest remaining elements into $\mathcal C$ so that the rest spans at most $N^{7/12}$ terms; each step bounds some $|s^\wedge\mathcal A|$ from below against Proposition 1. Not reconstructed here.

Dependencies

Freiman's inverse theorem (Proposition 3.1; Foundations of a Structural Theory of Set Addition, AMS Translations of Mathematical Monographs 37 (1973), Thm. 1.9, p. 11, and The addition of finite sets, Izv. Vyssh. Uchebn. Zaved. Mat. 1959, both not held), and Straus's bound $(4/\sqrt3+o(1))\sqrt N$ (J. Math. Sci. 1 (1966), 77--80, not held; reproved as Lemme 2 of the 1991 paper) for the standing upper bound.

Bears on

• Problem 874: the structure theorem on which the problem's status-defining result rests. The 1999 sequel quotes it as its Theorem 2 and derives from it, through its Proposition 1 and Theorem 3, the exact bound $\operatorname{card}\mathcal A\le2\sqrt{N+1/4}-1$ for $N\ge N_0$ (Theorem 1 of 1999), which gives $k(N)=\lfloor2\sqrt{N+1/4}-1\rfloor$ for large $N$. Within this paper it yields Theorem 1, $k(N)\le2N^{1/2}+CN^{5/12}$.