Bounded Exponential Sums with Multiplicative Coefficients

Technical Summary: Bounded Exponential Sums with Multiplicative Coefficients

Problem Statement
The paper investigates the conditions under which the exponential sum
$$S_f(x, \alpha) := \sum_{n \le x} f(n)e(n\alpha)$$
remains bounded as $x \to \infty$, where $f: \mathbb{N} \to \mathbb{C}$ is a multiplicative function, $\alpha \in \mathbb{R}$ is fixed, and $e(t) = e^{2\pi i t}$. While classical results (e.g., Daboussi, Montgomery-Vaughan) establish that $S_f(x, \alpha) = o(x)$ for irrational $\alpha$ when $f$ is bounded, this work focuses on the extreme case of boundedness ($\sup_x |S_f(x, \alpha)| < \infty$). The authors aim to classify the multiplicative functions $f$ for which this boundedness occurs, particularly in relation to the "pretentious" distance between $f$ and structured objects like twisted Dirichlet characters $\chi(n)n^{it}$.

Methodology
The paper employs a combination of pretentious number theory, sieve methods, and harmonic analysis. The methodology proceeds in three main stages:

1. Reduction to Pretentious Functions:
   The authors first establish that if $S_f(x, \alpha)$ is bounded, $f$ must be "pretentious," meaning it is close to a function of the form $\chi(n)n^{it}$ in the Granville-Soundararajan pretentious metric. This relies on Tao's logarithmically averaged Elliott conjecture and a van der Corput-type argument (Lemma 2.8) to show that bounded partial sums imply non-trivial self-correlation, which forces $f$ to resemble a twisted character.

2. The Rotation Trick:
   To refine the classification, the authors adapt the "rotation trick" introduced by Klurman, Mangerel, Pohoata, and Teräväinen (KMPT21). Originally designed for completely multiplicative functions, the authors extend this technique to general multiplicative functions (which may have zeros and higher prime power values). By constructing specific arithmetic progressions and utilizing concentration inequalities for pretentious functions (Lemma 3.1), they demonstrate that if $f$ deviates significantly from $\chi(n)n^{it}$ on a set of primes with divergent reciprocal sum, the exponential sum cannot remain bounded.

3. Large Sieve and Diophantine Approximation:
   For results depending on $\alpha$ (specifically Theorems 1.2 and 1.3), the authors utilize a large sieve lower bound for exponential sums (Theorem 4.2) derived from the first author's previous work. This allows them to relate the boundedness of $S_f(x, \alpha)$ to the distribution of rational approximations of $\alpha$. The proof invokes the Duffin–Schaeffer–Koukoulopoulos–Maynard theorem to handle cases where the sum is bounded for a set of $\alpha$ with positive measure.

Key Results

• Theorem 1.1 (General Classification):
  If $f$ is a multiplicative function with $f(n) \in \mathbb{U} \cup \{0\}$ (where $\mathbb{U}$ is the unit circle) and $\sum_{p:f(p)=0} 1/p < \infty$, and if $S_f(x, \alpha)$ is bounded for some $\alpha \in \mathbb{R}$, then there exists a Dirichlet character $\chi$ and a real number $t$ such that:
  $$\sum_p \max_k |f(p^k) - \chi(p^k)p^{ikt}| < \infty.$$
  This implies $f$ must be very close to $\chi(n)n^{it}$ on prime powers.

• Theorem 1.2 (Special Cases):
  For specific classes of functions, the classification is sharper:

  • If $f$ is completely multiplicative and $\alpha$ is irrational, boundedness implies $f(p) = \chi(p)p^{it}$ for all primes $p$.
  • If $f$ takes finitely many values, boundedness implies $f(p^k) = f(p^{k-1})\chi(p)$ for all but finitely many prime powers.
  • If $f$ is completely multiplicative and takes finitely many values, boundedness implies $f(p) = \chi(p)$ for all $p$ (i.e., $t=0$).

• Theorem 1.3 (Positive Measure Set):
  If the sum is bounded for a set of $\alpha$ with positive Lebesgue measure, the condition strengthens to:
  $$\sum_p \sum_k |f(p^k) - f(p^{k-1})\chi(p)p^{ikt}| < \infty.$$
  This result relies on the Duffin–Schaeffer theorem to show that the "distance" between $f$ and the twisted character must vanish in a specific $L^1$ sense over the primes.

• Sharpness (Section 5):
  The authors demonstrate that Theorem 1.2 does not hold for general multiplicative functions (only completely multiplicative ones) by constructing counterexamples where $f(p^k) = 1 + O(p^{-2k})$. They also define a "degree of pretentiousness" $\sigma(f)$ to show that the conditions in Theorems 1.1 and 1.3 are close to optimal.

Significance and Claims
The paper claims to provide a complete classification of multiplicative functions yielding bounded exponential sums in the case of completely multiplicative functions with irrational $\alpha$. It extends the scope of the "pretentious" framework to handle general multiplicative functions with zeros and higher prime power values, a significant technical hurdle addressed via the generalized rotation trick.

The work connects the Erdős Discrepancy Problem (specifically Tao's solution and Frantzikinakis's weighted variants) to the structure of multiplicative functions. It confirms that the only way to achieve boundedness in these sums is if the multiplicative function essentially mimics a Dirichlet character twisted by a power of $n$. The authors note that their results resolve a conjecture communicated by Cédric Pilatte (Conjecture 1.8) in the special case where $f$ is multiplicative and takes values in $\{0, 1\}$.

The paper does not propose new experimental applications but rather solidifies the theoretical understanding of the interplay between additive characters and multiplicative structures, refining the boundary between "random" multiplicative functions (which exhibit cancellation) and "structured" ones (which can sustain bounded sums).