🔢 mathematics

Better than square-root cancellation in Piatetski-Shapiro sequences

This paper establishes that Piatetski-Shapiro sequences exhibit better-than-square-root cancellation for sums of random multiplicative functions and achieve Weil's bound for character sums, thereby extending Harper's and Xu's recent results to a new class of sets with higher multiplicative energy.

Original authors: Renjie Zhu, Tianping Zhang

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Renjie Zhu, Tianping Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study how numbers behave when they are added or multiplied together. A central question in this field concerns the concept of "cancellation." Imagine a long list of numbers, some positive and some negative, that are added up. If the numbers are truly random, the positive and negative values tend to cancel each other out, leaving a final sum that is much smaller than the total count of numbers involved. For centuries, experts believed that this cancellation could never be better than a specific limit, known as the square-root barrier. This barrier suggests that if you add up a million numbers, the result will likely be around one thousand, not zero. This limit is so fundamental that it is tied to some of the most famous unsolved problems in mathematics, including the Riemann Hypothesis, which deals with the distribution of prime numbers.

However, recent discoveries have shaken this belief. Mathematicians have found that in certain special cases, the numbers cancel each other out even more effectively than the square-root limit predicts. This phenomenon, called "better than square-root cancellation," was first proven to exist for the set of all natural numbers and later for specific types of numbers like rough numbers. The big question remaining was whether this extra efficiency could happen in other, more sparse collections of numbers that do not share the same rich multiplication patterns as the natural numbers. If it could, it would suggest that the ability to cancel out is not dependent on a complex internal structure, but perhaps on something more subtle.

A team of researchers has now answered this question by turning their attention to a specific, sparse collection of numbers known as a Piatetski-Shapiro sequence. These sequences are created by taking a simple formula involving a power and rounding the result to the nearest whole number. While they look somewhat random, they are much thinner than the set of all natural numbers. The researchers investigated what happens when they add up random values assigned to these specific numbers. They found that, surprisingly, these sequences also exhibit the "better than square-root cancellation" effect. This means that even though the numbers in the sequence are sparse and lack the dense multiplication properties of natural numbers, the random values assigned to them still cancel each other out with exceptional efficiency.

To reach this conclusion, the authors had to navigate a significant mathematical hurdle. Previous methods used to prove this effect for natural numbers relied heavily on the fact that those numbers have a very specific, rich structure when multiplied together. The Piatetski-Shapiro sequences do not have this same structure, so the old tools did not work. Instead, the researchers developed a new approach that focused on the unique way these sequences are spaced out. They used a technique to smooth out the irregularities in the sequence, allowing them to isolate the random values and measure their behavior. By carefully analyzing the differences between consecutive terms in the sequence, they were able to show that the random values cancel out more than expected, proving that the phenomenon is not limited to sets with complex multiplicative structures.

The findings have immediate implications for another area of mathematics involving character sums, which are a way of measuring how numbers behave under different modular rules. The researchers showed that for almost all cases, the sums of these characters over the Piatetski-Shapiro sequences are much smaller than the standard limits would suggest. In fact, these sums reach a theoretical minimum bound known as Weil's bound for nearly every case. This result is significant because it confirms that the "better than square-root cancellation" is a robust phenomenon that can appear in diverse mathematical settings, not just in the most familiar number systems.

The study also clarifies what is not required for this extra cancellation to occur. The researchers noted that while the Piatetski-Shapiro sequences are different from natural numbers, they are not the same as another type of sequence called Beatty sequences. When they tried to apply their methods to Beatty sequences, they found that the extra cancellation does not always happen there. This distinction is important because it suggests that there is a specific, critical threshold of structure or spacing that allows for this efficient cancellation, but the exact nature of that threshold remains a mystery. The work does not solve the problem of exactly what makes a set capable of this behavior, but it successfully identifies a new, unexpected example where it does occur.

Ultimately, this paper expands our understanding of how randomness and structure interact in mathematics. It demonstrates that the ability of numbers to cancel each other out efficiently is not solely the domain of the most complex number sets. By proving that this phenomenon exists in the Piatetski-Shapiro sequences, the authors have provided a new piece of evidence that helps map the boundaries of this mathematical behavior. The result stands as a concrete proof that the square-root barrier is not an absolute wall, but a limit that can be surpassed in more varied circumstances than previously thought, opening the door for further exploration into the hidden patterns of sparse number sets.

Technical Summary: Better than Square-Root Cancellation in Piatetski-Shapiro Sequences

Problem Statement
The paper investigates the phenomenon of "better than square-root cancellation" in the partial sums of random multiplicative functions and Dirichlet character sums over Piatetski-Shapiro sequences. Specifically, the authors analyze the expectation of the absolute value of the sum $\sum_{n \in A} f(n)$ , where $A = N^{(c)}(x) = \{ \lfloor n^c \rfloor : n \le x \}$ is a Piatetski-Shapiro sequence with $1 < c < 2$ , and $f(n)$ is a Steinhaus or Rademacher random multiplicative function.

While it is well-established that for the set of natural integers $\mathbb{N}$ , Harper (2019) proved that $\mathbb{E}|\sum_{n \le x} f(n)| \asymp \sqrt{x}(\log \log x)^{-1/4}$ , which is strictly smaller than the standard square-root bound $\sqrt{x}$ , the behavior over sparse sets is less understood. Previous results by Soundararajan and Xu [13] suggested that if a set $A$ has "small" multiplicative energy (specifically $(2+o(1))|A|^2$ ), the sums exhibit standard square-root cancellation. Conversely, Harper (2019) and Xu (2023) showed that sets with specific structures (like natural numbers or $R$ -rough numbers) can achieve better than square-root cancellation. The central question addressed here is whether this phenomenon persists for Piatetski-Shapiro sequences, which possess a multiplicative energy larger than that of rough numbers but are still sparse, and whether similar bounds hold for character sums.

Methodology
The authors do not follow the martingale-based or random Euler product approaches used by Harper and Xu. Instead, the proof relies on the unique analytic structure of Piatetski-Shapiro sequences. The methodology proceeds as follows:

Indicator Function Decomposition: The sum over the sequence $N^{(c)}(x)$ is converted into a sum over all integers $n \le x$ using the indicator function $\chi^{(c)}(n)$ . Using Lemma 2.1, this indicator is approximated by a main term $\gamma n^{\gamma-1}$ (where $\gamma = 1/c$ ) and error terms involving the fractional part function $\psi(x)$ .
Vaaler's Approximation: The fractional part function $\psi(x)$ is expanded using Vaaler's trigonometric approximation (Lemma 2.2), transforming the problem into estimating exponential sums with random coefficients of the form $\sum f(n) (e(-h(n+1)\gamma) - e(-hn\gamma))$ .
Taylor Expansion and Cancellation: A critical step involves applying a Taylor expansion to the difference of exponential terms. The authors show that $e(-h(n+1)\gamma) - e(-hn\gamma)$ can be approximated by a term proportional to $n^{\gamma-1}$ . This factor $n^{\gamma-1}$ provides the necessary extra decay to avoid the difficult estimation of $\mathbb{E}|\sum e(hn\gamma)f(n)|$ , which is known to only exhibit square-root cancellation for certain irrational frequencies.
Cauchy-Schwarz and Orthogonality: The remaining expectations are bounded using the Cauchy-Schwarz inequality and the orthogonality properties of random multiplicative functions ( $\mathbb{E}[f(m)\overline{f(n)}] = \delta_{m,n}$ ). This reduces the problem to estimating sums of the form $\sum n^{2\gamma-2}$ , which are manageable given the range $1/2 < \gamma < 1$ .
Character Sums: For Dirichlet characters, the authors adapt the proof of Theorem 1.1 by replacing the expectation over random multiplicative functions with the mean value over all characters modulo a prime $p$ , utilizing a large sieve-type inequality established by Harper (2023).

Key Results
The paper establishes two main theorems:

Theorem 1.1: For a Steinhaus or Rademacher random multiplicative function $f(n)$ and a Piatetski-Shapiro sequence $N^{(c)}(x)$ with $1 < c < 2$ , the expected absolute sum satisfies:
$\mathbb{E} \left| \sum_{n \in N^{(c)}(x)} f(n) \right| = o\left( \sqrt{|N^{(c)}(x)|} \right).$
This confirms the existence of better than square-root cancellation for this class of sparse sets.
Theorem 1.2: For a large prime $p$ and Dirichlet characters $\chi \pmod p$ , the average size of character sums over the sequence is bounded by:
$\frac{1}{p-1} \sum_{\chi \pmod p} \left| \sum_{n \in N^{(c)}(x)} \chi(n) \right| \ll \frac{\sqrt{|N^{(c)}(x)|}}{(\log \log L)^{1/4}},$
where $L = \min\{|N^{(c)}(x)|, p/|N^{(c)}(x)|\}$ .
Corollary 1.3: As a consequence, for almost all characters (specifically, all but $o(p)$ exceptions), the character sum over the sequence satisfies the bound $\ll \sqrt{|N^{(c)}(x)|}$ , effectively reaching Weil's bound for almost all characters.

Significance and Claims
The authors claim that their result provides a "positive example" of better than square-root cancellation occurring in a set that does not have a multiplicative energy as small as $(2+o(1))|A|^2$ . Since Piatetski-Shapiro sequences have a different multiplicative structure compared to natural numbers or rough numbers (and their multiplicative energy is not sufficiently small to trivially yield the result), this suggests that a "rich" multiplicative structure is not a necessary condition for this phenomenon.

The paper modestly notes that while they have proven the existence of this phenomenon for Piatetski-Shapiro sequences, characterizing the exact "critical size" of multiplicative energy required for better than square-root cancellation remains an open problem. Furthermore, the authors discuss that their method does not immediately extend to Beatty sequences, as the specific cancellation mechanism relying on the Taylor expansion of the Piatetski-Shapiro indicator does not apply in the same way to Beatty sequences, where the behavior may depend on the Diophantine properties of the slope.

More like this