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Sárközy's theorem for shifted primes with restricted digits

This paper proves that any subset of natural numbers with positive upper Banach density contains two elements differing by a shifted prime with restricted digits, establishing this result by demonstrating the van der Corput property for such primes via local approximants to exponential sums.

Original authors: Alex Burgin

Published 2026-08-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Burgin

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

The Hidden Patterns in Number Prisons

Imagine the world of numbers as a giant, infinite city. Most people think of this city as a chaotic mess, but mathematicians have long suspected it's actually built on hidden, rhythmic patterns. One of the most famous neighborhoods in this city is the "Prime District," a place where numbers like 2, 3, 5, 7, and 11 live. These numbers are special because they can't be divided evenly by any other number except 1 and themselves. For decades, mathematicians have been trying to figure out how these primes are scattered. Do they follow a strict schedule, or are they just wandering aimlessly?

To make sense of this, mathematicians use a concept called "recurrence." Think of it like a game of musical chairs. If you have a large group of people (a set of numbers) and you start moving them around based on a specific rule (like adding a prime number), will you eventually find two people sitting in chairs that are exactly the right distance apart? A famous result from the 1970s by a mathematician named Sárközy proved that if you look at "shifted primes" (primes minus one), you will always find these matching pairs, no matter how you pick your group of people, as long as the group is big enough. This is a bit like saying that no matter how you arrange a crowd, if you ask everyone to step forward by a prime number, two people will inevitably land on the same spot.

But what happens if we put a strict rule on which primes we are allowed to use? Imagine a prison where only prisoners with specific tattoos are allowed to walk the halls. In the world of numbers, this is called "restricted digits." It means we only look at numbers that, when written in a certain base (like our usual base-10), only use a specific set of digits. For example, in base-10, we might only allow numbers made of the digits 0, 1, and 2. These numbers are very sparse; they are like islands in a vast ocean. The big question is: Do the "prison rules" break the rhythm? Does the musical chairs game still work if we only let the tattooed primes play?

The Paper's Discovery: Finding the Beat in a Sparse Crowd

In this paper, Alex Burgin tackles exactly that question. The author asks: If we restrict our primes to only those with specific digits (like only using the digits 0 and 1 in a very large base), do they still form a "recurrence set"? In other words, if we have a large collection of numbers, can we always find two of them that differ by one of these special, restricted primes minus one?

The answer, according to the paper, is a resounding yes. Burgin proves that even with these rigid digit restrictions, the shifted primes still act as a "recurrence set." This means that if you have a set of natural numbers that is "dense" enough (meaning it has a positive upper Banach density, which is a fancy way of saying it's not too scattered), you are guaranteed to find two numbers in that set, $a_1$ and $a_2$ , and a special prime $p$ (with restricted digits) such that $a_1 + p - 1 = a_2$ .

To reach this conclusion, the paper builds a sophisticated mathematical machine. The author doesn't just guess; they construct a "local approximant," which is like building a detailed map of the local neighborhood to understand the behavior of the primes. They use tools from Fourier analysis (a way of breaking complex waves into simple sine waves) to study the "exponential sums" of these primes. Think of these sums as listening to the "music" of the primes. If the music is chaotic and random, the primes are scattered. If the music has a specific rhythm, the primes are structured.

Burgin shows that these restricted primes have a very specific property called the "van der Corput property." This is a stronger version of recurrence. It's like proving that not only will two people land on the same chair, but the entire group will eventually align in a perfect pattern. The paper demonstrates that the "music" of these restricted primes cancels out in a very specific way at irrational frequencies (frequencies that don't repeat in a simple cycle). This cancellation is the key that unlocks the proof.

The paper explicitly rules out the idea that these results are just a lucky accident or a simple extension of previous work. The author notes that you cannot simply take the old rules for normal primes and apply them to these restricted ones, nor can you just take the rules for restricted numbers and apply them to primes. The combination is unique and requires a new approach. The paper also clarifies that while other mathematicians have looked at how these primes are distributed in specific arithmetic progressions (like primes that leave a remainder of 1 when divided by 3), this paper goes further by proving the "pointwise" behavior at every irrational frequency. This is a crucial distinction: it's not just about averages; it's about the behavior at every single specific point.

The confidence level here is high. The paper presents a rigorous proof, not a simulation or a suggestion. It uses a "transference principle," which is a method of borrowing a known result (that normal primes form a recurrence set) and adapting it to the new, restricted setting. The author carefully constructs the necessary mathematical bridges to show that the "digit restrictions" don't break the underlying rhythm of the primes.

One interesting detail the paper highlights is the necessity of the rules. The author points out that if you don't allow the digits 0 and 1 in your restricted set, the whole thing falls apart. For example, if you only allow even digits, you might never find a prime that fits the pattern. The paper proves that having 0 and 1 available is essential for the "prison" to still allow the musical chairs game to work.

In summary, this paper is a triumph of number theory. It takes a complex, sparse subset of the prime numbers—those with restricted digits—and proves that they still hold the deep, rhythmic secrets of the number world. It shows that even when you lock the primes in a digital cage, they still dance to the same beat as the free primes, ensuring that patterns and connections remain hidden within the chaos, waiting to be found.

Technical Summary: Sárközy's Theorem for Shifted Primes with Restricted Digits

Problem Statement
This paper investigates the additive recurrence properties of the set of shifted primes with restricted digits, denoted as $P_C - 1$ , where $P_C$ is the set of primes whose base- $b$ expansions consist solely of digits from a specific subset $A \subset \{0, \dots, b-1\}$ . The central question is whether the classical recurrence phenomenon established by Sárközy (that the set of shifted primes $P-1$ is a set of recurrence) persists when the primes are "thinned" by a rigid digit restriction.

The author addresses whether, for any subset $A \subset \mathbb{N}$ with positive upper Banach density, there exist distinct elements $a_1, a_2 \in A$ and a restricted-digit prime $p$ such that $a_1 + p - 1 = a_2$ . This problem is non-trivial because recurrence and the van der Corput property are not generally inherited by arbitrary subsets; the sparsity and additive structure of the restricted digit set $C$ complicate the analysis of the associated exponential sums.

Methodology
The proof strategy relies on establishing the van der Corput property for the set $P_C - 1$ . Following the framework of Kamae and Mendès France, this requires constructing a sequence of trigonometric polynomials supported on $P_C - 1$ that approximate the indicator function of the rationals $\mathbb{Q}/\mathbb{Z}$ in the limit.

The technical core of the paper involves a detailed analysis of exponential sums over the restricted digit set $C$ and the von Mangoldt function $\Lambda(n)$ . The methodology proceeds through several key stages:

Local Approximants and Major Arcs: The author constructs a local approximant for the exponential sums $\sum 1_C(n)\Lambda(n)e(n\theta)$ . They utilize a Dirichlet-type theorem for fixed moduli to derive an asymptotic formula for the number of restricted-digit primes in arithmetic progressions. This involves separating the sum into "major arcs" (rational frequencies) and "minor arcs" (irrational frequencies).
Fourier Estimates:
- Major Arcs: The author proves an asymptotic formula (Theorem 1.2.1) for the count of restricted-digit primes in arithmetic progressions $n \equiv t \pmod m$ . This formula explicitly separates the standard prime-to- $b$ congruence obstruction from a finite $b$ -adic digit density.
- Minor Arcs: The paper establishes a Vinogradov-type equidistribution theorem (Theorem 1.2.2), showing that for any irrational $\theta$ , the exponential sum $\sum_{n < b^N} 1_C(n)\Lambda(n)e(n\theta)$ is $o((b-s)^N)$ .
- Hybrid Estimates: To control the error terms, the author adapts techniques from Maynard's work on primes with restricted digits. They employ $L^1$ , Large Sieve, and Hybrid estimates (Lemmas 3.1.2, 3.1.6, 3.1.9) to bound the exponential sums over the restricted digit set $C$ . A crucial component is the "Inversion with Few Frequencies" (Proposition 3.3.1), which allows the control of the Fourier transform of $C$ using a small number of frequencies.
Spectral Argument: Using the established Fourier decay on irrational frequencies and the precise asymptotic on rational frequencies, the author constructs the necessary trigonometric polynomials. They demonstrate that the limit of these polynomials is the indicator function of the rationals, thereby satisfying the criteria for the van der Corput property.

Key Results
The paper establishes two primary theorems under specific "standing conditions" on the digit set $A$ (requiring $A$ to contain $\{0, 1\}$ , be a union of intervals, and have sufficient density relative to the base $b$ ):

Theorem 1.2.1 (Dirichlet-type Theorem): For any fixed modulus $m$ and residue $t$ , the number of restricted-digit primes $p < b^N$ with $p \equiv t \pmod m$ is asymptotically $\kappa_{m,t}(b-s)^N$ , where $\kappa_{m,t}$ is an explicit constant accounting for local obstructions.
Theorem 1.2.2 (Irrational Decay): For any irrational $\theta$ , the exponential sum over restricted-digit primes satisfies $\sum_{n < b^N} 1_C(n)\Lambda(n)e(n\theta) = o((b-s)^N)$ .
Theorem 1.3.1 (Sárközy-type Result): If $A \subset \mathbb{N}$ has positive upper Banach density, there exists a prime $p \in P_C$ and distinct $a_1, a_2 \in A$ such that $a_1 + p - 1 = a_2$ .

Significance and Claims
The author positions their work as complementary to recent results by Nath and Leng–Sawhney. While Nath provides stronger distribution estimates (Bombieri–Vinogradov type) for primes omitting a single digit in large bases, and Leng–Sawhney prove a ternary additive theorem, neither directly yields the pointwise irrational-frequency decay required to establish the van der Corput property for shifted restricted-digit primes.

The paper claims novelty in:

Combining Local and Global Theories: Successfully integrating an explicit fixed-modulus local theory with irrational Fourier decay.
Generality: Extending these results to a broader class of digit sets (unions of intervals) rather than just single missing digits.
Recurrence: Proving that the shifted restricted-digit primes form a van der Corput set, a strictly stronger notion of recurrence than simple intersectivity.

The author explicitly notes that their methods do not currently extend to proving the existence of finite IP sets of arbitrary dimension within $P_C - 1$ (which would require higher-order Gowers norm estimates, $U^k$ for $k \ge 3$ ), nor do they claim to improve the level of distribution for the primes themselves beyond what is necessary for the recurrence application. The primary contribution is the establishment of the van der Corput property for this specific sparse set of primes.

The Hidden Patterns in Number Prisons

The Paper's Discovery: Finding the Beat in a Sparse Crowd

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