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Source. Lemma 5.4 (p. 347), with the notation of §5.2 (pp. 346–347) and Lemma 5.3 (p. 347), of G. Grekos, L. Haddad, C. Helou, J. Pihko, On the Erdős–Turán conjecture, Journal of Number Theory 102 (2003), no. 2, 339–352, the edition named on the source card.

Statement

Setting (§5.2). $A,B\subseteq\mathbb N$ (with $\mathbb N=\{0,1,\ldots\}$), $d\in\mathbb N$, and

$$C=A+d*B=\{a+db:a\in A,\ b\in B\},$$

where $A$ is finite and $d>\max A$. As before, $r(P,n)=|\{(p,q)\in P\times P:p+q=n\}|$ counts ordered pairs, and $f_P(X)=\sum_{p\in P}X^p$ (§4.1, p. 346).

Lemma 5.3 (p. 347). $f_C(X)=f_A(X)f_B(X^d)$. Its proof shows that the translates $a+d*B$, $a\in A$, are pairwise disjoint.

Lemma 5.4 (p. 347). For $n\in\mathbb N$ write $n=dq+e$ with $q,e\in\mathbb N$ and $0\le e<d$. Then

$$r(C,n)=r(A,e)\,r(B,q)+r(A,d+e)\,r(B,q-1).$$

When $q=0$ the second term is absent: the proof sums $r(A,j)r(B,k)$ over $k\in\mathbb N$ only, so $r(B,-1)$ is read as $0$ (a reading made here; the paper defines $r(B,n)$ for $n\in\mathbb N$).

Read depth. Claims checked: the setting, Lemma 5.3 and Lemma 5.4 were read clause by clause on the printed pp. 346–347, and the short proofs of both were read.

Proof pointer

Squaring Lemma 5.3 gives $g_C(X)=g_A(X)g_B(X^d)$ for the generating series $g_P=f_P^2$ of $r(P,\cdot)$, so $r(C,n)=\sum r(A,j)r(B,k)$ over $j+dk=n$. Since $r(A,j)=0$ for $j>2\max A$ and $2\max A<2d$, only $j=e$ ($k=q$) and $j=d+e$ ($k=q-1$) survive.

The paper uses the lemma for Proposition 5.6 and Theorem 5.7.

Bears on

• Problem 1145: an identity for the self-representation function of one set; it gives no bound on the problem's cross count $1_A*1_B$.