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Source. Theorem 2, p. 3, of Sándor Z. Kiss and Csaba Sándor, Generalized Sidon sets of perfect powers, The Ramanujan Journal 59 (2022), no. 2, 351--363, doi:10.1007/s11139-022-00622-z. Labels and pages are those of arXiv:2006.02783v1 (4 June 2020), the edition named on the source card.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the printed page. The proof (Section 3, pp. 5--8) was read for structure only. Nothing here is independently reviewed.

Statement

Setting (pp. 1--2). For $h\ge2$ and an infinite set $A$ of positive integers, $R_{A,h}(n)$ is the number of solutions of $a_1+\cdots+a_h=n$ with $a_1<a_2<\cdots<a_h$ in $A$, and $R^*_{A,h}(n)$ the number with $a_1\le a_2\le\cdots\le a_h$; $A$ is a $B_h[g]$ set when $R^*_{A,h}(n)\le g$ for every positive integer $n$. $A(n)$ counts the members of $A$ up to $n$, and $(\mathbb Z^+)^k=\{1^k,2^k,3^k,\ldots\}$.

Theorem 2 (p. 3). Let $k$ be a positive integer. Suppose that for some $2\le h\le k$ and for every $\eta>0$ there is a positive integer $n_0(\eta)$ with $R_{(\mathbb Z^+)^k,h}(n)<n^\eta$ for every $n\ge n_0(\eta)$. Then for every $\varepsilon>0$ there is a set $A\subseteq(\mathbb Z^+)^k$ such that $R_{A,h}(n)$ is bounded and

$$A(x)\gg x^{\frac1k-\varepsilon}=x^{\min\{\frac1k,\frac1h\}-\varepsilon}.$$

The hypothesis is of the kind Hardy and Littlewood's Hypothesis K asserts for $h=k$; the paper notes (p. 3) that Hypothesis K holds for $k=2$ and fails for $k=3$ (Mahler). The theorem bounds the strict-order count $R_{A,h}$, which is weaker than the $B_h[g]$ condition for $h\ge3$; the paper's closing remark (p. 11) records the passage to $B_h[g]$ sets as not achieved. For $h=2$ a bounded $R_{A,2}$ gives a $B_2[g]$ set for some $g$, which is Corollary 1.

Proof pointer

Section 3 (pp. 5--8). Lemma 6 (p. 5) shows that a random set whose expected $R_{A,l}(n)$ is $\ll n^{-\varepsilon}$ for every $2\le l\le h$ has $R_{A,h}(n)$ bounded with probability 1, by the Erdős-Tetali disjointness lemma and the Erdős-Rado sunflower lemma. The hypothesis is first transferred from $h$ to every $2\le l\le h$ (p. 7); then each $k$-th power $n$ is taken independently with probability $n^{-\varepsilon}$ (p. 8), and Lemma 7 (p. 6), a Chernoff bound with Borel-Cantelli, gives the density for $\varepsilon<1/k$.

Dependencies

Lemmas 1--5 of the paper (pp. 4--5), cited from the literature: the Erdős-Rényi probability space, Borel-Cantelli, the Erdős-Tetali disjointness lemma, a Chernoff inequality and the Erdős-Rado $\Delta$-system lemma.

Bears on

• Problem 158: none directly. The theorem bounds representations by a constant it does not specify, so it produces no $B_2[2]$ set; the paper does not mention the problem.