Subconvexity of Short $k$-Free Exponential Sums

Technical Summary: Subconvexity of Short k-Free Exponential Sums

Problem Statement
This paper investigates the moments of short exponential sums over $k$-free integers. Let $k \ge 2$ be an integer, and let $\mu_k(n)$ denote the indicator function for $k$-free integers (integers not divisible by any $k$-th power of a prime). The central object of study is the exponential sum over a short interval $(N-K, N]$:
$$S_k(\alpha; K) = \sum_{N-K < n \le N} \mu_k(n) e(n\alpha),$$
where $e(x) = e^{2\pi i x}$. The primary goal is to determine the range of interval lengths $K$ (relative to $N$) for which the $L^s$-moments of this sum satisfy the "expected" asymptotic behavior known for the full interval case ($K=N$). Specifically, the paper seeks to establish bounds of the form:
$$\int_0^1 |S_k(\alpha; K)|^s d\alpha \asymp_{k,s} \begin{cases} K^{\frac{s}{k+1}} & \text{if } s < 1 + \frac{1}{k}, \\ K^{s-1} & \text{if } s > 1 + \frac{1}{k}. \end{cases}$$
The paper defines an exponent $\theta_{k,s}$ as the infimum of all $\theta$ such that these bounds hold for any $K \gg N^\theta$. The objective is to lower the value of $\theta_{k,s}$, thereby proving that the expected behavior persists in shorter intervals, specifically breaking the "square-root barrier" ($\theta < 1/2$).

Methodology
The approach relies on a decomposition of the exponential sum into "middle part" sums and the application of a refined $L^2$-estimate for these components.

1. Decomposition: The sum $S_k(\alpha; K)$ is decomposed using the identity $\mu_k(n) = \sum_{d^k|n} \mu(d)$. The author defines "middle part" sums $c_n(y, z) = \sum_{y < d \le z, d^k|n} \mu(d)$ and partitions the exponential sum into dyadic segments $T_i(\alpha)$ based on the size of $d$. The total sum is split into a "low" part $h_D(\alpha)$ (summing small $d$) and a "high" part $H_D(\alpha)$ (summing large $d$), where $D \approx \log K$.
2. The Core Estimate (Lemma 1.2): The paper's technical engine is a new bound on the mean square of the middle part sums:
   $$\sum_{N-K < n \le N} |c_n(y, z)|^2 \ll K y^{1-k} + N^\varepsilon K^{1/k} + N^{\delta_k + \varepsilon},$$
   where $\delta_k$ is a specific constant depending on $k$. This estimate combines several tools:
   • Perron's Formula: Used to handle the multiplicative structure when the summation range is small.
   • Hyperbola Method: Applied to shorten long sums involving the sawtooth function $\psi(x)$.
   • Van der Corput's Method: Both single and multi-dimensional versions are utilized. The paper employs the resolution of the main conjecture of Vinogradov's Mean Value Theorem to select optimal exponent pairs $(p, q)$ for these estimates.
   • Zeta Function Bounds: Standard convexity bounds for $\zeta(s)$ are used in conjunction with Dirichlet polynomials.
3. Moment Interpolation: The bounds for the $L^s$-moments are derived by interpolating between the $L^1$ and $L^2$ estimates of the decomposed sums. For the subcritical case ($s < 1 + 1/k$), the argument uses Fejér kernels to smooth the sums and handle the lower bound. For the critical and supercritical cases ($s \ge 1 + 1/k$), the paper utilizes Hölder's inequality and careful pointwise estimates of the high-part sums $H_D(\alpha)$.

Key Contributions and Results

• Theorem 1.1 (Main Result): The paper establishes explicit upper bounds for the admissible exponent $\theta_{k,s}$. For $s \neq 1 + 1/k$, the paper proves:
  $$\theta_{k,s} \le \begin{cases} \left(\frac{k+1}{2}\right)\delta_k & \text{if } s < 1 + \frac{1}{k}, \\ \left(\frac{1}{2} + \frac{1}{2(s-1)}\right)\delta_k & \text{if } 1 + \frac{1}{k} < s < 2, \\ \delta_k & \text{if } s \ge 2. \end{cases}$$
  Here, $\delta_2 = 105/317 \approx 0.331$ and for $k \ge 3$, $\delta_k$ is defined by a specific rational function of $k$ that is strictly less than $1/(k+1)$.
• Breaking the Square-Root Barrier: A significant consequence is that for $s=1$ (the $L^1$-mean), the expected behavior holds for intervals of length $K \gg N^\theta$ where $\theta < 1/2$. Specifically, for $k=4$, the bound reaches $K \gg N^{1555/3142 + \varepsilon} \approx N^{0.49491}$.
• Theorem 1.5 (Critical Moment): For the critical exponent $s = 1 + 1/k$, the paper proves:
  $$K^{1/k} \log K \ll \int_0^1 |S_k(\alpha; K)|^{1+1/k} d\alpha \ll K^{1/k} (\log K)^{2-1/k},$$
  valid for $K \gg N^{(\frac{k+1}{2})\delta_k + \varepsilon}$. This improves the upper bound for the full interval case ($K=N$) slightly.
• Corollary 1.8 (Möbius Function Application): By relating the $k$-free sum to the Möbius-twisted sum via orthogonality, the paper derives a lower bound for the $L^1$-mean of $\sum \mu(n)e(n\alpha)$. It shows that for $K \gg N^{105/317 + \varepsilon} \approx N^{0.49685}$, the integral is $\gg K^{1/6}$. This improves upon the previous best result by Sun ($K \gg N^{9/17}$).

Significance and Claims
The paper claims that the strength of the bound in Lemma 1.2 is the "sole impediment" to achieving $\theta_{k,s} = 0$. The results demonstrate that further improvements to the $L^2$-estimate of the middle part sums (specifically reducing the exponent $\Delta_k$ in equation 1.5) would immediately translate to stronger subconvexity results for all moments.

The author notes that the case $s=2$ corresponds to the classical problem of counting $k$-free integers in short intervals. While the paper's bounds for this specific case do not yet match the strongest known results (e.g., Filaseta and Trifonov for $k$-free gaps), the methodology provides a unified framework. The paper explicitly states that the improvement in the $L^1$-mean for the Möbius function is a direct consequence of the refined Lemma 1.2.

The work is presented as a generalization and improvement of previous results by Sun, Balog, Ruzsa, and Keil, utilizing modern tools from the theory of exponential sums (specifically the resolution of Vinogradov's Mean Value Theorem) to push the limits of subconvexity in short intervals. The paper does not claim to solve the gap problem for $k$-free integers completely but suggests that the derived bounds on $\theta_{k,2}$ are a step toward better gap results.