Large sieve inequalities with power moduli and Waring’s problem

Let $M$ be an integer, let $N$ be a positive integer, and let $\{z_{n}\}$ be a sequence of complex numbers. Let $\mathcal{S}$ be a finite set of real numbers, and let $\delta\leq 1$ be a positive real number. We write $\lVert\alpha\rVert$ for the distance from a real number $\alpha$ to its nearest integer, and we also write $e(\alpha)=e^{2\pi i\alpha}$. Moreover, we say that the set $\mathcal{S}$ is $\delta$-spaced modulo 1 if $\lVert\alpha-\beta\rVert\geq\delta$ whenever $\alpha$ and $\beta$ are distinct members of $\mathcal{S}$. The general large sieve inequality says that if $\mathcal{S}$ is $\delta$-spaced modulo 1, then

$\displaystyle\sum_{\alpha\in\mathcal{S}}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq(\delta^{-1}+N-1)\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$

(1)

Montgomery wrote a lovely expository article on this result [Mon78].

Let $Q$ and $k$ be positive integers. Then we take the set

$\displaystyle\mathcal{S}_{k}(Q)=\left\{\text{$a/q^{k}:a$ and $q$ are coprime positive integers with $a\leq q^{k}\leq Q^{k}$}\right\}$

(2)

so that

$$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}=\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}.$$

This paper is about large sieve inequalities with $k\textsuperscript{th}$-power moduli. These are inequalities of the form

$$\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}\leq\Delta_{k}(Q,N)\sum_{n=M+1}^{M+N}|z_{n}|^{2},$$

where $\Delta_{k}(Q,N)$ is independent of $M$ and $\{z_{n}\}$. We now let $\Delta_{k}(Q,N)$ be the infimum of such constants so that our large sieve inequalities are just upper bounds on $\Delta_{k}(Q,N)$.

In the first paper on these bounds, Zhao observed two bounds that trivially follow from the general large sieve inequality [Zha04]. Firstly, since the set $\mathcal{S}_{k}(Q)$ is $Q^{-2k}$-spaced modulo 1, we have

$\displaystyle\Delta_{k}(Q,N)\leq Q^{2k}+N-1.$

(3)

Secondly, we can apply the general large sieve inequality to the sum over $a$, and then sum over $q$. This gives

$\displaystyle\Delta_{k}(Q,N)\leq\sum_{q=1}^{Q}(q^{k}+N-1)\leq Q(Q^{k}+N-1).$

(4)

If $N\gg Q^{2k}$, then bound (3) gives $\Delta_{k}(Q,N)\ll N$. If $N\ll Q^{k}$, then bound (4) gives $\Delta_{k}(Q,N)\ll Q^{k+1}$. Combining these, we have

$\displaystyle N\gg Q^{2k}\mbox{ or }N\ll Q^{k}\implies\Delta_{k}(Q,N)\ll Q^{k+1}+N.$

(5)

It’s not hard to see that

$\displaystyle\Delta_{k}(Q,N)\gg Q^{k+1}+N.$

(6)

So, in particular, (5) is sharp. If $z_{M+1}=1$ and every other $z_{n}=0$, then we compute that

$$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}=|\mathcal{S}_{k}(Q)|\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$$

Therefore

$\displaystyle\Delta_{k}(Q,N)\geq|\mathcal{S}_{k}(Q)|=\sum_{q=1}^{Q}\varphi(q^{k})=\sum_{q=1}^{Q}q^{k-1}\varphi(q)\gg Q^{k+1}.$

(7)

If each $z_{n}=e(-n\alpha_{0})$, for a fixed $\alpha_{0}\in\mathcal{S}_{k}(Q)$, then we have

$$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\geq\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha_{0})\right|^{2}=N\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$$

Therefore

$\displaystyle\Delta_{k}(Q,N)\geq N.$

(8)

Combining bounds (7) and (8), we get bound (6).

In the case $k=1$, bounds (3) and (6) collectively show that $\Delta_{1}(Q,N)\asymp Q^{2}+N$. Otherwise, we call

$\displaystyle Q^{k}<N<Q^{2k}$

(9)

the nontrivial range. This is where many authors [Zha04, BZ05, Bai06, BZ08, Hal12, Hal15, Hal18, Hal20, Mun21, BMS22, McG22] beat the trivial bounds and worked toward Zhao’s conjecture that

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}(Q^{k+1}+N).$

Baier, Lynch and Zhao showed that the factor of $Q^{\varepsilon}$ cannot be removed when $k=2$ [BLZ19]. Their method extends to all $k\geq 2$.

Our method of attacking the nontrivial range is simple, effective and can easily be applied to more general large sieve inequalities with polynomial moduli. We choose to illustrate our method in this restricted setting of $k\textsuperscript{th}$-power moduli for three reasons. Firstly, these large sieve inequalities have received a lot of attention, and we achieve savings in substantial ranges for all $k\geq 4$. Secondly, they have many (actualised and potential) applications, as listed in the introduction to [BMS22]. Lastly, our method relates these large sieve inequalities to Waring’s problem with $k\textsuperscript{th}$-powers.

For an introduction to Vinogradov’s mean value theorem and its application to Waring’s problem, look to Vaughan’s book [Vau97, Ch. 5]. In the case $k=3$, Wooley proved a long standing conjectured bound on Vinogradov mean values [Woo16, Thm. 1.1]. Bourgain, Demeter and Guth were the first to prove the conjectured bound for all $k\geq 3$ [BDG16, Thm. 1.1]. We make use of Wooley’s generalised version [Woo18, Thm. 1.1] of Bourgain, Demeter and Guth’s bound. See [Coo+22] for an interesting comparison between the method of Wooley and that of Bourgain, Demeter and Guth.

Let $Q,N,n$ and $k,s,t$ be positive integers. In light of the introduction, we restrict to the nontrivial range

$\displaystyle Q^{k}<N<Q^{2k}.$

Let $R_{k,s}(n)$ be the number of $s$-tuples of positive integers $(n_{1},\ldots,n_{s})$ satisfying $n=n_{1}^{k}+\ldots+n_{s}^{k}$. The forthcoming implied constants are independent of $Q,N,n$. They may depend on $k,s,t$ or on an arbitrarily small positive real number $\varepsilon$.

We pass to counting Farey fractions in short intervals in Section 4. This is standard practice. We then lay the foundations of our new method in Section 5, where we also prove the following two lemmata.

In Section 6, we recall an elementary divisor bound for $R_{k,2}(n)$. Running this through Lemma 2.1 with $s=2$ recovers Baier and Zhao’s large sieve inequality [BZ05, Thm. 1.1]. We state this large sieve inequality in part (i) of our first theorem.

Also in Section 6, we recall Marmon’s non-elementary bound on $R_{k,4}(n)$ [Mar10, Thm. 1.3]. Marc Munsch suggested that we insert this into Lemma 2.1 with $s=4$ so as to recover Baker, Munsch and Shparlinski’s large sieve inequality [BMS22, Thm. 1.4]. We state this large sieve inequality in part (ii) of our first theorem. In fact, we state the inequality for all $k\geq 3$, while Baker, Munsch and Shparlinski left a void when $k=4$. McGrath [McG22, Cor. 1.4] filled this void, albeit with a weaker bound than the general formula in $k$ would suggest. Our result corrects this, filling the void with the appropriate bound.

Finally, in Section 7, we invoke a recent and high-powered theorem of Wooley [Woo18, Thm. 1.1]. We end up with a family, indexed by positive integers $t\leq k$, of mean value estimates of the form (10). Running this through Lemma 2.2 with $s=\frac{1}{2}t(t+1)$ results in our second theorem.

As we’ll see in the next section, Theorem 2.4 with $t=k$ is particularly powerful.

To discuss a conditional result, we assume that $k\geq 3$ and $s\geq k+1$ throughout the rest of this section. Recall the Hardy-Littlewood singular series

$\displaystyle\mathfrak{S}_{k,s}(n)=\sum_{q=1}^{\infty}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q}\left(\frac{1}{q}\sum_{r=1}^{q}e(ar^{k}/q)\right)^{s}e(-na/q).$

Then Hardy and Littlewood proved that whenever $s$ is sufficiently large relative to $k$, we have the formula

$\displaystyle R_{k,s}(n)=\frac{\Gamma(1+\frac{1}{k})^{s}}{\Gamma(\frac{s}{k})}\mathfrak{S}_{k,s}(n)n^{\frac{s}{k}-1}+o(n^{\frac{s}{k}-1}).$

(12)

See [Vau97, Ch. 2] for a modern proof of this formula, with $s>2^{k}$ and a power saving in the error term. We are often interested in obtaining $\mathfrak{S}_{k,s}(n)\gg 1$ so that the first term on the right-hand side is not swallowed by the error, thereby making (12) a fully-fledged asymptotic formula. This lower-bound on the singular series can be subtle, as covered in [Vau97, §. 4.5]. Fortunately, for our application, the only bound we need on the singular series is a modest upper-bound. Indeed, we have [Vau97, Thm. 4.3, p. 49]

$$\mathfrak{S}_{k,s}(n)\ll n^{\varepsilon}.$$

Therefore, if formula (12) holds, then we have

$\displaystyle R_{k,s}(n)\ll n^{\frac{s}{k}-1+\varepsilon}.$

(13)

If bound (13) holds, then it follows from Lemma 2.1 that

$\displaystyle\Delta_{k}(Q,N)\ll N^{1-\frac{1}{s}}Q^{1+\frac{k}{s}+\varepsilon}.$

(14)

Rewriting this large sieve bound as $NQ^{1+\varepsilon}(N^{-1}Q^{k})^{1/s}$ and recalling that $N>Q^{k}$, it’s optimal to choose $s$ as small as possible. As luck would have it, this runs with – not against – the difficulty in Waring’s problem! The Hardy-Littlewood asymptotic formula (12) is conjectured to hold with $s=k+1$ whenever $k\geq 3$. If this famous conjecture holds, then large sieve bound (14) holds with $s=k+1$ whenever $k\geq 3$.

Let $Q,N,k$ be positive integers. The forthcoming implied constants are independent of $Q$ and $N$, but they may depend on $k$ or on an arbitrarily small positive real number $\varepsilon$.

The standard way of comparing bounds is to write $N=Q^{\lambda}$ for a real parameter $\lambda$. Restricting to the nontrivial range (9), we assume that $k<\lambda<2k$. It is also customary to introduce clean notation for some functions of $k$ that appear in exponents. Put

$\displaystyle\kappa=\frac{1}{2^{k-1}}\qquad\mbox{and}\qquad\omega=\frac{1}{(k-1)(k-2)+2}.$

Let $Q,N$ and $k,s,t$ be positive integers. We also let $M$ be an integer, and let $\{z_{n}\}$ be a sequence of complex numbers. The forthcoming implied constants are independent of $Q,M,N$ and $\{z_{n}\}$. But they may depend on $k,s,t$ or on an arbitrarily small positive real number $\varepsilon$. We assume that

$\displaystyle Q^{k}\leq N\ll Q^{2k}.$

Dividing the summation over $q$ into dyadic intervals gives

$\displaystyle\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}\ll Q^{\varepsilon}\sup_{1\leq x\leq Q}\sum_{\frac{x}{2}<q\leq x}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}.$

Let $x$ be a real number in the interval $1\leq x\leq Q$ and write

$$\mathcal{A}_{k}(x):=\left\{\frac{a}{q^{k}}:1\leq a\leq q^{k},\ \frac{x}{2}<q\leq x\ \mbox{and}\,\gcd(a,q)=1\right\}$$

so that

$$\sum_{\frac{x}{2}<q\leq x}\sum_{\begin{subarray}{c}a=1\\ \gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}=\sum_{\alpha\in\mathcal{A}_{k}(x)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}.$$

We want to hit this with the general large sieve inequality (1), leading us to consider the modulo 1 spacing of the set $\mathcal{A}_{k}(x)$. Let $\delta=\delta(x)\leq x^{-k}$ be a positive real number. Note that $\min\mathcal{A}_{k}(x)\geq x^{-k}$ and that $\max\mathcal{A}_{k}(x)\leq 1$. Then, for every $\alpha,\beta\in\mathcal{A}_{k}(x)$, we have $\lVert\alpha-\beta\rVert<\delta$ if and only if $|\alpha-\beta|<\delta$. Hence, for every $\alpha_{0}\in\mathcal{A}_{k}(x)$, we have

$$\mathcal{A}_{k}(x,\delta,\alpha_{0}):=\{\alpha\in\mathcal{A}_{k}(x):\lVert\alpha-\alpha_{0}\rVert<\delta\}=\{\alpha\in\mathcal{A}_{k}(x):|\alpha-\alpha_{0}|<\delta\}.$$

Now observe that there exists a partition of the set $\mathcal{A}_{k}(x)$ into

$$\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|$$

parts so that each part is $\delta$-spaced modulo 1. Thus, for any one of these parts $\mathcal{P}$, say, the general large sieve inequality (1) gives

$$\sum_{\alpha\in\mathcal{P}}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq(N-1+\delta^{-1})\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$$

This then yields the bound

$$\sum_{\alpha\in\mathcal{A}_{k}(x)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq\left(\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\right)(N-1+\delta^{-1})\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$$

Putting everything together, we may take

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}\sup_{1\leq x\leq Q}\ \inf_{0<\delta\leq x^{-k}}\left((N+\delta^{-1})\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\right).$

Recall that $x^{k}\leq Q^{k}\leq N$. Therefore $\frac{1}{2N}\leq\frac{1}{2x^{k}}$, and so we may take

$$\delta=\frac{1}{2N}$$

in the infimum. Hence, with this choice of $\delta$, we have

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}N\sup_{1\leq x\leq Q}\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|.$

(23)

We keep the notation of the previous section. Recall that $x$ is a real number in the interval $1\leq x\leq Q$ and that we have

$$\delta=\frac{1}{2N}\leq\frac{1}{2x^{k}}.$$

Let $\alpha_{0}\in\mathcal{A}_{k}(x)$. Then we write $a_{0}$ and $q_{0}$ for the positive integers satisfying

$$\alpha_{0}=\frac{a_{0}}{q_{0}^{k}},\qquad a_{0}\leq q_{0}^{k},\qquad\frac{x}{2}<q_{0}\leq x\qquad\mbox{and}\qquad\gcd(a_{0},q_{0})=1.$$

Let $\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$ be the set of positive integers $q$ in the dyadic interval $\frac{x}{2}<q\leq x$ for which there exists an integer $b$ satisfying $|b|<\delta x^{2k}$ and $a_{0}q^{k}\equiv b\bmod q_{0}^{k}$.

Given any $\alpha\in\mathcal{A}_{k}(x,\delta,\alpha_{0})$, we write $\alpha=a/q^{k}$ in reduced form like we did for $\alpha_{0}$. Then we rewrite the condition $|\alpha-\alpha_{0}|<\delta$ as $|a_{0}q^{k}-aq_{0}^{k}|<\delta q^{k}q_{0}^{k}\leq\delta x^{2k}$. Therefore $\alpha\mapsto q$ defines a function

$$\mathcal{A}_{k}(x,\delta,\alpha_{0})\hookrightarrow\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0}).$$

Since $\delta$ is sufficiently small, this function is injective. To see this, take two fractions $\alpha,\beta\in\mathcal{A}_{k}(x,\delta,\alpha_{0})$ with the same reduced denominator $q^{k}$. Now observe that $|\alpha-\beta|\leq|\alpha-\alpha_{0}|+|\alpha_{0}-\beta|<2\delta$. Then we see that $\alpha=\beta$, because otherwise we would get the contradictory bound $|\alpha-\beta|\geq q^{-k}\geq x^{-k}\geq 2\delta$. Hence

$\displaystyle|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\leq|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|.$

(24)

Now we let $\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$ be the set of positive integers $q$ in the interval $s(\frac{x}{2})^{k}<q\leq sx^{k}$ for which there exists an integer $b$ satisfying

$\displaystyle|b|<s\delta x^{2k}\qquad\mbox{and}\qquad a_{0}q\equiv b\bmod q_{0}^{k}.$

(25)

Evidently, if $q_{1},\ldots,q_{s}\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$, then $q_{1}^{k}+\ldots+q_{s}^{k}\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$. This leads us to consider, for each $q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, the number $R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)$ of $s$-tuples $(q_{1},\ldots,q_{s})\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})^{s}$ satisfying $q=q_{1}^{k}+\ldots+q_{s}^{k}$. We have the formula

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}=\sum_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q).$

We proceed in two ways. First, we take the supremum to get

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}\leq|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\sup_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q).$

(26)

Second, we use Cauchy-Schwarz to get

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{2s}\leq|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\sum_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)^{2}.$

(27)

Either way, we need a bound on $|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|$.

To count the number of $q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, we first recall condition (25), from which we see that there are $O(1+x^{2k}\delta)$ choices for $a_{0}q\bmod q_{0}^{k}$. Then, because $\gcd(a_{0},q_{0}^{k})=1$, there are $O(1+x^{2k}\delta)$ choices for $q\bmod q_{0}^{k}$. Note that $q\asymp x^{k}\asymp q_{0}^{k}$. Thus, for a fixed residue $r$, there are $O(1)$ choices for $q$ with $q\equiv r\bmod q_{0}^{k}$. Putting these estimates together, we have

$\displaystyle|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\ll 1+x^{2k}\delta\ll N^{-1}Q^{2k}.$

(28)