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Claim. Kurt Mahler, Note on Hypothesis K of Hardy and Littlewood, starts from the identity

$$(9\xi^4)^3+(3\xi-9\xi^4)^3+(1-9\xi^3)^3=1$$

and replaces $\xi$ by $\xi/\eta$, which gives $(9\xi^4)^3+(3\xi\eta^3-9\xi^4)^3+(\eta^4-9\xi^3\eta)^3=\eta^{12}$. All three cubes are positive when $\eta>0$ and $0<\xi<9^{-1/3}\eta$, so each such integer $\xi$ writes $\eta^{12}$ as a sum of three positive cubes, and distinct $\xi$ give distinct triples. Mahler concludes that for all large $N$ which are twelfth powers, $x^3+y^3+z^3=N$ has at least $9^{-1/3}N^{1/12}$ solutions in non-negative integers, which refutes Hardy and Littlewood's Hypothesis K for $n=3$. In the notation of Problem 322, with $A$ the cubes, $1_A^{(3)}(n)\ge9^{-1/3}n^{1/12}$ for infinitely many $n$, so for every $c<1/12$ there are infinitely many $n$ with $1_A^{(3)}(n)>n^c$.

Covers. The second question for $k=3$: yes. The order of growth of $1_A^{(3)}(n)$, and every $k\ge4$, are not addressed; Mahler's identities in $n$ variables have terms of both signs and give no lower bound for sums of $n$ non-negative $n$th powers.

Depends on. No page of this wiki.

Acceptance. Refereed: J. London Math. Soc. 11 (1936), no. 2, 136--138 (received 10 December 1935). The Crossref record gives the issue as April 1936 and no day, so the page is dated to the first day of that month. The site's commentary credits the result to Mahler, but on a problem the site labels OPEN that commentary is not review. The paper's library card records the theorem.