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Statement

Setting (pp. 115--116). Write $\rho>0$ in base $2$ as $\rho=\sum_{i=-k}^{\infty}\varepsilon_i(\rho)2^{-i}$ with $\varepsilon_i(\rho)\in\{0,1\}$ and $\varepsilon_i(\rho)=0$ infinitely often. $\rho$ is an infinite dyadic fraction (IDF) if $\varepsilon_i(\rho)=1$ infinitely often, and a finite dyadic fraction (FDF) otherwise; the FDF are thus the positive dyadic rationals. Definition 1 (p. 115) gives $A_{\alpha\beta}$ type $\mathcal F$ when $\alpha$ and $\beta$ are both FDF, type $\mathcal M$ when $\alpha$ is FDF and $\beta$ is IDF, and type $\mathcal I$ when both are IDF. $A_{\alpha\beta}$, $P$ and completeness are as in Theorem 1.

Definition 2 (p. 116). For $\alpha,\gamma$ FDF, $m^*_\gamma=\min\{m\mid\varepsilon_k(\gamma)=0\text{ for all }k>m\}$, the place of the last nonzero binary digit of $\gamma$, and

$$g_\alpha(m)=\bigl|\{\beta\mid m^*_\beta=m\text{ and }A_{\alpha\beta}\text{ is complete}\}\bigr|/2^m,$$

where the set is printed with "$B$" in place of $\beta$. The paper says that $g_\alpha(m)$ counts the $\beta$ for which $A_{\alpha\beta}$ is complete and of type $\mathcal F$; it does not state over which $\beta$ the count runs, and its proof (p. 119) takes the number of FDF $\beta$ with $m^*_\beta=j$ to be $2^j$. The proof writes $j^*$ for $m^*$.

Theorem 2 (p. 116, quoted). "Let $\alpha>0$ be FDF. Then $\lim_{m\to\infty}g_\alpha(m)=1$."

The proof gives the rate $g_\alpha(j)>1-j^{A+1}/2^j$ with $A=2\,[2^{j^*_\alpha}\alpha]$ (p. 119).

Source. N. Hegyvári, On sumset of certain sets, Publ. Math. Debrecen 45 (1994), no. 1--2, 115--122: the definitions on pp. 115--116, the statement on p. 116, the proof in Section 3, pp. 118--119.

Read depth. Claims checked: the definitions and the statement were read clause by clause on the journal print. The proof was read but not checked step by step.

Proof pointer

Section 3, pp. 118--119. The proof rests on a lemma printed as Lemma 2 on p. 119 (the paper also labels an earlier lemma, p. 117, Lemma 2), described as a quantitative form of a result of the author's 1989 paper: for a positive integer $m$ and a nonnegative integer $s$, if $\sum_{i\ge1}\varepsilon_i(\beta)>2m^2$ then some element of $P(A_\beta)$ is congruent to $s$ modulo $m$. With $A=2\,[2^{j^*_\alpha}\alpha]$, a $\beta$ with $\sum_{i\ge1}\varepsilon_i(\beta)>A$ gives a complete $A_{\alpha\beta}$, so the incomplete $\beta$ with $j^*_\beta=j$ number at most $\sum_{n=1}^{A}\binom jn<j^{A+1}$.

Bears on

• Problem 354: a pair of type $\mathcal F$ has both $\alpha$ and $\beta$ dyadic rationals, so $\alpha/\beta$ is rational and outside the problem's hypothesis. The theorem decides no case of the problem; it concerns the rational-ratio pairs the problem excludes.