Average sizes of mixed character sums

Technical Summary of "Average Sizes of Mixed Character Sums"

Problem Statement
The paper investigates the average size of mixed character sums of the form:
$$S(\chi) = \sum_{1 \le n \le x} \chi(n) e(n\theta) w(n/x)$$
where $\chi$ is a Dirichlet character modulo a large prime $r$, $\theta$ is a real number, $w$ is a smooth weight function supported on $[0,1]$, and $x \le r$. The authors aim to determine the typical magnitude of this sum when averaged over all characters $\chi \pmod r$.

Previous work by Harper established that for rational $\theta$ (or in the random multiplicative function model), the average size is $o(\sqrt{x})$, exhibiting "better than square-root cancellation." However, the behavior for irrational $\theta$ remained an open question. The central problem addressed is whether the average size remains $o(\sqrt{x})$ or reverts to the "square-root size" ($\asymp \sqrt{x}$) when $\theta$ is irrational and satisfies a specific Diophantine condition.

Methodology
The authors employ a moment method, specifically analyzing the second and fourth moments of the sum to bound the first absolute moment via Hölder's inequality. The proof is divided into two regimes based on the size of $x$ relative to $r$:

1. Short Sum Case ($x \le \sqrt{r}$):

   • The authors utilize the orthogonality of Dirichlet characters. In this range, the congruence condition $m_1 m_2 \equiv n_1 n_2 \pmod r$ in the fourth moment expansion simplifies to an equality $m_1 m_2 = n_1 n_2$ because the variables are small compared to $r$.
   • The problem reduces to bounding an exponential sum over integer solutions to $m_1 m_2 = n_1 n_2$.
   • The authors adapt techniques from Heap and Soundararajan (specifically [13] in the bibliography) regarding random multiplicative functions. They parametrize the solutions and use Möbius inversion to detect coprimality.
   • A key technical step involves a Diophantine condition on $\theta$ (specifically $\|q\theta\| \ge C \exp(-q^{1/4})$) to bound exponential sums where the frequency is non-zero. They define a set $L$ of "bad" integers and show that the contribution from terms where the frequency lies in $L$ is negligible due to the spacing properties of $L$.

2. Long Sum Case ($\sqrt{r} \le x \le r$):

   • Direct orthogonality fails as the variables exceed $\sqrt{r}$. Instead, the authors apply the Poisson summation formula to transform the sum over $n \in [1, x]$ into a dual sum over frequencies $m$ modulo $r$, effectively replacing the range $x$ with $r/x$.
   • This transformation introduces a dual problem involving the Fourier coefficients of the weight function and Gauss sums.
   • The fourth moment computation reduces to counting the number of solutions to a specific quadratic Diophantine equation involving congruences:
     $$k(m_1 + m_2 - n_1 - n_2) \equiv n_1 n_2 - m_1 m_2 \pmod r$$
     where $k = \lfloor r\theta \rfloor$.
   • The authors utilize an injection lemma (Lemma 3.2) to map the 4-variable solution set to a 3-variable set defined by a hyperbolic equation $ab + 2cS = S^2 - 4P$.
   • The counting problem is then solved using the pigeonhole principle combined with the Diophantine approximation properties of $\theta$. They show that the number of solutions $(S, P)$ to the congruence $kS \equiv P \pmod r$ within the relevant ranges is sufficiently small, preventing the fourth moment from growing too large.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The authors prove that for an irrational $\theta$ satisfying the weak Diophantine condition $\|q\theta\| \ge C \exp(-q^{1/4})$, the average size of the mixed character sum is of order $\sqrt{x}$. Specifically:
  $$\sqrt{x} \ll \frac{1}{r-1} \sum_{\chi \pmod r} \left| \sum_{1 \le n \le x} \chi(n) e(n\theta) w(n/x) \right| \ll \sqrt{x}$$
  This holds uniformly for all $x \le r$.
• Contrast with Rational $\theta$: The result confirms that the "better than square-root cancellation" phenomenon observed by Harper for rational $\theta$ (and random models) does not persist for typical irrational $\theta$. The additive twist $e(n\theta)$ destroys the delicate structure required for extra cancellation when $\theta$ is irrational.
• Diophantine Condition: The paper establishes that a relatively weak Diophantine condition (weaker than those required for random multiplicative functions in previous literature) is sufficient to ensure the square-root size. This condition is satisfied by most irrational numbers, including $\pi$, $e$, and algebraic irrationals.
• Distribution Question: The authors raise the question of whether the normalized sum converges to a standard complex Gaussian distribution. They note that while this holds for small $x$ (where orthogonality is perfect), recent work by Bober, Klurman, and Shala suggests the answer is negative for the full range $x=r$.

Significance
The paper demonstrates that the "better than square-root cancellation" in character sums is a fragile property, highly sensitive to the arithmetic nature of the additive twist. By proving that the average size returns to the standard $\sqrt{x}$ bound for irrational $\theta$, the authors delineate the boundary between the random-like behavior of rational twists and the deterministic behavior of irrational twists. The work connects the study of mixed character sums to the theory of critical multiplicative chaos and Diophantine approximation, providing a rigorous framework for understanding how additive perturbations affect the distribution of multiplicative functions. The technical reduction of the fourth moment to a counting problem for quadratic Diophantine equations represents a novel methodological contribution to the field.