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OPEN

Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that
\[1_A^{(k)}(n) >n^c?\]

Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\leq n^{o(1)}$, but this was disproved by Mahler [Ma36] for $k=3$, who constructed infinitely many $n$ such that
\[1_A^{(3)}(n)\gg n^{1/12}\]
(where $A$ is the set of cubes). Erdős believed Hypothesis $K$ fails for all $k\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\epsilon>0$
\[\sum_{n\leq N}1_A^{(k)}(n)^2 \ll_\epsilon N^{1+\epsilon}.\]
Erdős and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.'

Independently Erdős [Er36] and Chowla proved that for all $k\geq 3$ and infinitely many $n$ \[1_A^{(k)}(n) \gg n^{c/\log\log n}\] for some constant $c>0$ (depending on $k$).

OPEN

Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that
\[f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}\]
for all $\epsilon>0$? Is it true that if $m<k$ then
\[f_{k,m}(x) \gg x^{m/k}\]
for sufficiently large $x$?

This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'. The case $k=2$ was resolved by Landau, who showed
\[f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}\]
for some constant $c>0$.

For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.

OPEN

Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that
\[f_{k,3}(x) \gg x^{3/k}\]
or even $\gg_\epsilon x^{3/k-\epsilon}$?