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Source. Theorem 3, 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 was read clause by clause on the printed page. The proof (Section 4, pp. 8--11) was read for structure only. Nothing here is independently reviewed.

Statement

Setting as in Theorem 2: $R_{A,h}(n)$ counts solutions of $a_1+\cdots+a_h=n$ with $a_1<\cdots<a_h$ in $A$.

Theorem 3 (p. 3). For every $k\ge2$ there is a positive integer $h_0(k)=O(8^kk^2)$ such that for every $h\ge h_0(k)$ and every $\varepsilon>0$ there is a set $A\subseteq(\mathbb Z^+)^k$ such that $R_{A,h}(n)$ is bounded and

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

The theorem is unconditional. As with Theorem 2, it bounds the strict-order count $R_{A,h}$, not $R^*_{A,h}$, so it does not give a $B_h[g]$ set; the paper's closing remark (p. 11) leaves that extension open.

Proof pointer

Section 4 (pp. 8--11). Each $k$-th power $n$ is taken independently with probability $n^{-(\frac1k-\frac1h+\varepsilon)}$. The expected $R_{A,l}(n)$ is shown to be $\ll n^{-\varepsilon}$ for every $2\le l\le h$: directly for $l\le h/k$, and for $h/k<l\le h$ by a dyadic decomposition and Lemma 8 (p. 9), a weaker form of a lemma of Vu bounding the number of solutions of $y_1^k+\cdots+y_l^k=n$ in boxes for $l\ge h_2(k)=O(8^kk^2)$. Lemma 6 (p. 5) then gives bounded $R_{A,h}$, and the expectation of $A(x)$ gives the density.

Dependencies

Lemma 6 of the paper (p. 5) and Lemma 8 (p. 9), the latter from V. H. Vu, On a refinement of Waring's problem, Duke Math. J. 105 (2000), Lemma 2.1.

Bears on

• Problem 158: none. The theorem concerns $h$-fold sums for large $h$, not sums of two elements; the paper does not mention the problem.