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Statement

Notation (p. 204): $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are strictly increasing sequences of positive integers and $A(N)$ counts the elements of $A$ up to $N$.

Tijdeman's conjecture (p. 217). R. Tijdeman conjectured, in a letter to the first author, that every infinite difference intersector set $B=\{b_1,b_2,\ldots\}$ satisfies

$$\liminf_{k\to\infty}\frac{b_{k+1}}{b_k}=1.\qquad(43)$$

The paper proves it as Theorem 7. A note added in proof (p. 217) reports that C. L. Stewart and R. Tijdeman had meanwhile proved the conjecture independently, unpublished.

Theorem 7 (p. 217, quoted with its displays). "If $\Delta>1$, $B=\{b_1,b_2,\ldots,b_k,\ldots\}$ is a strictly increasing infinite sequence of positive integers and"

$$\inf_{k=1,2,\ldots}\frac{b_{k+1}}{b_k}\ge\Delta\;(>1),\qquad(44)$$

"then there exists a strictly increasing sequence $A=\{a_1,a_2,\ldots,a_k,\ldots\}$ of positive integers such that"

$$\liminf_{n\to\infty}\ \text{[sic]}\ \frac{A(N)}{N}\ge\exp\Bigl(-\Bigl(\frac{\log3}{\log\Delta}+1\Bigr)\log24\Bigr)\qquad(45)$$

"and the equations"

$$a_x-a_y=b_z,\qquad(46)$$ $$a_u+a_v=b_t\qquad(47)$$

"are not solvable."

The limit in (45) is printed with $n\to\infty$ under $\liminf$ and $N$ in the quotient, marked [sic]; it is the limit as $N\to\infty$. The one sequence $A$ avoids both (46) and (47), and (47) allows $u=v$. Since such an $A$ has positive lower density, a $B$ satisfying (44) is neither a difference nor a sum intersector set, which is (43).

Best possibility (pp. 222--223, Section 6). For difference intersector sets the paper states that Theorem 7 is best possible: a union $B=\bigcup_i\{n_i,n_i+1,\ldots,n_i+j_i\}$ with $n_i\to+\infty$ rapidly and $j_i\to+\infty$ slowly is a difference intersector set by Theorem 5, and has $b_{k+1}/b_k>1+\varepsilon_k$ (62) with $\varepsilon_k\to0$ arbitrarily slowly. For sum intersector sets the paper does not know, and asks as question (ii) (p. 223): is it true that if $\varepsilon_k\to0$ (and $\varepsilon_k>0$) then there is an infinite sequence $B$ such that (62) holds and $\liminf_{N\to+\infty}A(N)/N>0$ implies the solvability of (47)?

Source. P. Erdős and A. Sárközy, On differences and sums of integers, II, Bull. Soc. Math. Grèce (N.S.) 18 (1977), no. 2, 204--223: the conjecture and the statement on p. 217, Lemma 1 on pp. 217--219, Lemma 2 on pp. 219--221, the completion of the proof on pp. 221--222, Section 6 on pp. 222--223. The edition read is identified on the source card.

Read depth. Claims checked: the conjecture, the statement, the note added in proof, the Section 6 remark and question (ii) were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 217--222. Lemma 1 (pp. 217--219): if $0<\gamma\le1$ and positive integers $d_1,d_2,\ldots$ satisfy $d_{k+1}/d_k\ge2+\gamma$, some real $\alpha$ has $\lVert d_k\alpha\rVert\ge\gamma/6$ for all $k$, by nested closed intervals. Lemma 2 (pp. 219--221): for $0<\delta<1$ and reals $\alpha_1,\ldots,\alpha_k$, more than $(\delta/2)^kN$ integers $n\le N$ have every $\lVert n\alpha_i\rVert<\delta$, for $N$ large, by pigeonhole. For Theorem 7, choose $k$ with $\Delta^k\ge3>\Delta^{k-1}$ (57) and split $B$ into the $k$ subsequences $\{b_i,b_{i+k},b_{i+2k},\ldots\}$, each with ratios at least $3$; Lemma 1 with $\gamma=1$ gives $\alpha_i$ with $\lVert b\alpha_i\rVert\ge\frac16$ on the $i$th subsequence (60). Let $A$ be the integers $a$ with every $\lVert a\alpha_i\rVert<\frac1{12}$ (61); Lemma 2 with $\delta=\frac1{12}$ gives (45), and any difference or sum of two elements of $A$ has $\lVert\cdot\,\alpha_i\rVert<\frac16$ for every $i$, so it is not in $B$.