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Statement

Setting (p. 292). For an integer $k\ge2$ and positive integers $M\le N$, $f_k(M,N)$ is the least size of a set $A\subset[0,M]$ such that every positive integer $n\le N$ is $a+b^k$ with $a\in A$ and $b$ a positive integer.

Theorem 3 (p. 293). Suppose $0<\delta<1$ is a fixed real number. There are constants $k_0=k_0(\delta)>2$ and $N_k(\delta)>2$ such that if $N\ge N_k(\delta)$ and $k\ge k_0$, then

$$f_k(\delta N,N)\ge C(\delta)\,\frac{k^\Delta}{\log k} \Bigl(1-\frac{\log(1+1/\delta)}{\log k}\Bigr)N^{1-1/k},$$

where

$$C(\delta)=\frac{\delta\,\Delta^\Delta\log(1+1/\delta)}{(1+\Delta)^{1+\Delta}}, \qquad \Delta=\frac{\log1/\delta}{\log(1+1/\delta)}.$$

Since $1<1/\delta<1+1/\delta$, the exponent satisfies $0<\Delta<1$ (an observation of this page), so for fixed $\delta$ the bound grows more slowly in $k$ than the constant $k$ of Theorem 1, which needs $\delta$ small in terms of $k$.

Source. Wenguang Zhai, The additive completion of $k$th powers, J. Number Theory 79 (1999), 292--300, doi:10.1006/jnth.1999.2441: the setting on p. 292, Theorem 3 on p. 293, the proof in Section 5 on p. 299. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the constants were read clause by clause on the printed pages. The proof is a sketch in the paper and was not checked; nothing here is independently reviewed.

Proof pointer

Section 5, p. 299. The derivative comparison of the proof of Theorem 1, with $\delta$ now fixed, gives (24); bounding the binomial sum by $\delta^d(1+k^{-1}(1+1/\delta)^d)$ when $k>2(1+1/\delta)^d$ gives the lower bound (26) for $\lvert A\rvert N^{1/k-1}$. The paper then chooses $d$ of order $\log k/\log(1+1/\delta)$, with a parameter $D$ printed as "$D=1+\Delta/\Delta$" [sic], and states that Theorem 3 follows from (26); the optimization is not written out.

Dependencies

The argument of Theorem 1 of the same paper, inequalities (6)--(16).

Bears on

• Problem 33: the theorem needs $k\ge k_0(\delta)>2$, so it does not cover the squares, $k=2$, that the problem concerns, and it decides neither of the problem's questions.