I am working on the Waring problem, essentially through Vaughan's book. I have an issue understanding a specific point. Consider the equation $$x_1^k + \cdots + x_s^k \equiv n \mod q \qquad (\star)$$

where $x_1$ and $q$ are coprime. Vaughan gives the following lemma:

Let $A$ a set of classes mod $q$ of cardinality $a$, $B$ a set of classes mod $q$ of cardinality $b$ such that $0 \in B$ and every $b \not\equiv 0$ (mod q) is coprime to $q$. Then $$|A+B| \geq \min(q, a+b-1)$$

The question now is : for $s$ large enough, does $(\star)$ have a solution? Vaughan claims that it is only a repeated application of this lemma, however I do not see how. I can imagine we should take $A=\{n\}$ and $B$ containing $k$-th power residues, however I am stuck with this. Could someone help me?

  • $\begingroup$ Well, on a very simple level, $(\star)$ always has a solution with $s \leq \min(q,n)$, each $x_i \in \{0,1\}$. So, yes, there is a solution for large enough $s$, where "large enough" is in terms of $q$ or $n$. ... Is $q$ prime? Do you assume $x_2,\dotsc,x_n$ are also coprime to $q$, or only just $x_1$? Are $q$ and $n$ fixed, or are you trying to find an $s$ that works for a fixed $q$ and all $n$, or an $s$ that works for fixed $n$ and all $q$...? $\endgroup$ – Zach Teitler Jun 12 at 18:05

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