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Given $\lambda>1$ and $d\in\mathbb N$, we consider the set $$\mathcal W(\lambda,d)=\{\sum_{0\leq x_1\leq x_2\leq \dots\leq x_d}x_1^\lambda+\ldots+x_d^\lambda\}$$ of all possible sums of (at most) $d$ integers to the power $\lambda$.

Question: Does there always exist a smallest integer $w(\lambda)$ such that $\mathcal W(\lambda,w(\lambda))$ has only bounded gaps?

If yes and if $\lambda\not\in\mathbb N$ does there exist an integer $w'(\lambda)\geq w(\lambda)$ such that $\mathcal W(\lambda,w'(\lambda))$ has only a finite number of gaps of size at least $\epsilon$ for any strictly positive $\epsilon$?

Easy observation: $w(\lambda)$ exists for all $\lambda>1$ in $\mathbb Q$ by the original proof for Waring numbers (for $\lambda=a/b$ consider only integers of the form $n^b$). Similarly, $w'(\lambda)$ exists for all $\lambda\in\mathbb Q\setminus\mathbb N$ larger than $1$.

Trivial observation on the original Waring problem: The definition of $w(\lambda)$ is also of limited interest for $\lambda$ integral: We have for example $w(2)=3$. Indeed, for any $k$ there are $k$ consecutive integers which are not sums of two squares: Consider $N+1,\ldots,N+k$ such that $N+j\equiv p_{i_j}\pmod{p_{i_j}^2}$ for $k$ distinct primes $p_{i_1},\ldots, p_{i_k}$ in $3+4\mathbb N$. On the other hand, the set $4^n(7+8m)$ of all integers which are not sums of three squares contains no consecutive elements. (The definition of $w'$ is of course meaningless for $\lambda\in\mathbb N$.)

Final remark: The question concerning $w(\lambda)$ is essentially equivalent to a question on integers: Round $x^\lambda$ to some integer at distance at most $B$ (with $B$ arbitrary but independent of $x$). If the set of all rounded integers is coprime, almost all natural integers are sums of a bounded number (given by a suitable increase of $w(\lambda)$) of such rounded powers.

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  • $\begingroup$ I like the question, but I’d replace $\mathcal{W}$ with $S$ (for the set of sums), since $w$ and $w’$ also appear in the question. $\endgroup$
    – user44143
    Oct 1, 2021 at 13:51
  • $\begingroup$ @MattF. I like to use uppercase calligraphic letters for sets and the corresponding ordinary lowercase letters for integers associated to these sets. (Puts less strain on memory requirements of my slightly outdated main processor.) $\endgroup$ Oct 1, 2021 at 14:55

1 Answer 1

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I believe this is the paper you want: https://arxiv.org/abs/2107.14536

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    $\begingroup$ Your answer would be much more useful if you added where exactly to look in that 31-pages long technical article. Link-only answers often get deleted from this site. $\endgroup$
    – Alex M.
    Oct 28, 2021 at 13:01
  • $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Oct 28, 2021 at 13:01
  • $\begingroup$ m.mathnet.ru/php/… $\endgroup$
    – alpoge
    Oct 28, 2021 at 15:37
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    $\begingroup$ Joking about also posting only a link and leaving aside (though it really does look like a pile-on above! —- presumably because of automatic processes), I’m not 100% confident in my reading of the above link, but I found it through the link posted in the answer (referenced in the introduction) and I think(??) it answers the question? I admit I might have read everything way too quickly… $\endgroup$
    – alpoge
    Oct 28, 2021 at 15:42

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