Duffin--Schaeffer examples, real residue systems, and Bohr-set primes

This paper generalizes the Duffin–Schaeffer theorem to real residue systems by establishing a zero-one law for inhomogeneous approximation based on the parameter's membership in specific sets, utilizing novel results on the distribution of primes in Bohr sets and the equidistribution of circle rotations.

The Gist

Here is an explanation of the paper "Duffin–Schaeffer Examples, Real Residue Systems, and Bohr-Set Primes" using simple language and creative analogies.

The Big Picture: A Game of "Almost"

Imagine you are trying to hit a moving target with a dart. In mathematics, this is called Diophantine approximation. You have a number (the target) and you want to see how closely you can get to it using fractions (the darts).

Usually, mathematicians ask: "Can I get arbitrarily close to this number using fractions?"

• Homogeneous case: The target is just a number $x$. You want to find a fraction $p/q$ such that $x$ is very close to $p/q$.
• Inhomogeneous case: The target is shifted. You want to find a fraction $p/q$ such that $x$ is very close to $p/q + y$ (where $y$ is a specific "offset" or shift).

For a long time, mathematicians knew a rule (the Khintchine Theorem) that said: "If the sum of your 'dart sizes' (how close you need to get) is infinite, you will hit the target almost everywhere. If the sum is finite, you will miss almost everywhere."

The Problem: This rule only worked if your "dart sizes" were shrinking smoothly (monotonically). In 1941, Duffin and Schaeffer found a trick: if you make the dart sizes jump around wildly (non-monotonic), the rule breaks. You can have an infinite sum of dart sizes, yet still miss the target completely for certain numbers.

What This Paper Does: The "Chameleon" Strategy

The authors of this paper (Hesseling, Ramírez, and Hauke) wanted to prove a super-powerful version of this trick. They asked:

"Can we design a single set of 'dart sizes' (a function $\psi$) that behaves like a chameleon?

• For a specific list of 'bad' targets ($Y$), the darts miss every single time (measure 0).
• For a different list of 'good' targets ($Z$), the darts hit almost everywhere (measure 1)."

They proved YES, we can do this, provided the "good" targets aren't just simple combinations of the "bad" ones.

The Three Magic Ingredients

To pull off this magic trick, the authors had to invent or extend three specific mathematical tools. Think of these as the ingredients in their recipe.

1. Real Residue Systems (The "Floating Fence")

Imagine you have a fence made of repeating sections (like a picket fence). In standard math, the gaps in the fence are always at whole number intervals (0, 1, 2...).

• The Innovation: The authors created a "Real Residue System." Imagine the fence sections can be shifted to any position on the ground, not just integer spots.
• The Result: They proved a rule (extending a theorem by Rogers) that says: "No matter how you shift these floating fence sections, the total area they cover is always at least as big as if they were lined up perfectly at zero."
• Why it matters: This guarantees that when they try to hit their "good" targets, the fence (the approximation set) is wide enough to catch them.

2. Bohr Sets (The "Vibe Check" Zones)

To build their "dart sizes," they needed to pick specific numbers (integers) to use as denominators. They couldn't just pick random numbers; they needed numbers that share a specific "vibe" or pattern.

• The Concept: A Bohr set is a collection of numbers that, when multiplied by a specific angle, land very close to a target spot on a circle.
• The Analogy: Imagine a group of runners on a circular track. A Bohr set is the group of runners who, at a specific moment, are all standing within a few inches of a specific finish line marker.
• The Challenge: They needed to know: "Do these specific groups of runners contain enough primes (special numbers) to make the math work?"

3. Primes in Bohr Sets (The "Prime Hunters")

This is where the paper gets very technical but also very beautiful. They had to prove two things about primes living inside these "Vibe Check" zones:

1. The Prime Number Theorem for Bohr Sets: They proved that if a zone is big enough, it contains infinitely many primes, and they are distributed in a predictable way (just like primes are distributed in standard arithmetic progressions).
2. Equidistribution: They proved that if you take these special primes and multiply them by a "good" target number, the results are scattered perfectly evenly around the circle.

• The Metaphor: Imagine you have a sieve (a filter) that only lets through numbers that fit a specific pattern (the Bohr set). The authors proved that if you run all the prime numbers through this sieve, the ones that come out are still perfectly random and evenly spread out. This randomness is crucial for ensuring the "good" targets get hit.

How the Proof Works (The Construction)

The authors didn't just say "it exists"; they built it step-by-step:

1. The List: They list all the "bad" targets ($Y$) and "good" targets ($Z$).
2. The Blocks: They create the function $\psi$ in blocks. Each block is designed to handle one specific "bad" target and one specific "good" target.
3. The Trap: For the "bad" target, they use the Real Residue System logic to show that the approximation sets are actually very small (they miss).
4. The Net: For the "good" target, they use the Prime in Bohr Set logic. Because the primes are so well-distributed, the approximation sets overlap in a way that covers almost the entire number line (they hit).
5. The Mix: They use a "mixing" technique (like stirring a pot) to ensure that the blocks don't interfere with each other. They space them out so far apart that the "misses" for the bad targets don't accidentally ruin the "hits" for the good targets.

The Appendix: A Surprising Twist

The paper includes an appendix by Manuel Hauke that answers a side question: "If you have a huge set of numbers, can you always find a smaller subset where the numbers don't share many common factors, but their reciprocals still add up to infinity?"

• The Answer: No.
• The Analogy: Imagine a giant library of books. You might think you can always find a shelf of books where no two share a common author, but the shelf is still "heavy" enough (infinite sum). Hauke proved you can construct a library so cleverly that any shelf you pick either shares too many authors or is too light.

Summary

This paper is a masterclass in controlled chaos.

• It takes a known mathematical rule that usually works smoothly.
• It breaks the smoothness to create a "monster" function.
• It uses advanced tools (floating fences, prime hunters, and patterned zones) to prove that this monster can be tamed to behave exactly how we want: ignoring specific numbers while embracing others.

It's like building a machine that can simultaneously ignore a specific group of people while welcoming everyone else, all based on the hidden patterns of prime numbers.

Technical Summary

Here is a detailed technical summary of the paper "Duffin–Schaeffer Examples, Real Residue Systems, and Bohr-Set Primes" by Stefan M. Hesseling and Felipe A. Ramírez (with an appendix by Manuel Hauke).

1. Problem Statement and Context

The paper addresses a fundamental problem in Diophantine approximation, specifically the inhomogeneous Duffin–Schaeffer conjecture and the construction of counterexamples to monotonicity assumptions in Khintchine-type theorems.

• Background: The classical Khintchine theorem (1924) states that for a non-increasing function $\psi: \mathbb{N} \to \mathbb{R}_{\ge 0}$, the set of $\psi$-approximable numbers $W(\psi, 0)$ has Lebesgue measure 0 or 1 depending on whether $\sum \psi(q)$ converges or diverges. The inhomogeneous version (Szüsz, 1958) extends this to $W(\psi, y) = \{x : \|qx - y\| < \psi(q) \text{ i.o.}\}$ for a fixed shift $y$.
• The Issue: Duffin and Schaeffer (1941) showed that the monotonicity of $\psi$ is essential for the divergence case in the homogeneous setting. Ramírez (2017) extended this to the inhomogeneous setting, showing that for any countable set $Y$, one can find a non-monotonic $\psi$ such that $\sum \psi(q)$ diverges, yet $\mu(W(\psi, y)) = 0$ for all $y \in Y$.
• The Gap: Ramírez's result for the inhomogeneous case (showing $\mu(W(\psi, z)) = 1$ for $z$ in a disjoint set $Z$) was conditional on a dynamical conjecture related to covering systems.
• Goal: The authors aim to prove an unconditional generalization: For any countable sets $Y$ and $Z$ (with $Z$ rationally independent of $Y \cup \{1\}$), construct a function $\psi$ such that $\mu(W(\psi, y)) = 0$ for $y \in Y$ and $\mu(W(\psi, z)) = 1$ for $z \in Z$, without relying on unproven conjectures.

2. Methodology

The proof relies on a sophisticated construction of the function $\psi$ supported on specific blocks of integers, combined with three major theoretical advancements:

1. Bohr Sets and Real Residue Systems:

   • The support of $\psi$ is constructed using products of primes lying in Bohr sets (sets of the form $N_v(y, \epsilon) = \{n \in \mathbb{N} : \|ny - v\| < \epsilon\}$).
   • The authors introduce Real Residue Systems, generalizing classical covering systems where residues $a_i$ can be arbitrary real numbers rather than integers. They define $R(\epsilon, a, n) = \bigcup ([a_i - \epsilon, a_i + \epsilon] + n_i \mathbb{Z})$.
   • They prove a generalization of Rogers's Theorem (1961), showing that the measure of a real residue system is minimized when the residues are zero ($a=0$). This allows them to bound the measure of approximation sets for $y \in Y$ by comparing them to the homogeneous case.

2. Distribution of Primes in Bohr Sets:

   • To ensure the constructed sets have the necessary density and independence properties, the authors must understand the distribution of primes within Bohr sets.
   • They generalize the Siegel–Walfisz Theorem (Prime Number Theorem for Arithmetic Progressions) to primes in Bohr sets.
   • They generalize Vinogradov's Theorem on the equidistribution of $(pz)_{p \in \mathbb{P}}$ modulo 1 to primes lying in Bohr sets, provided the rotation angle $z$ is rationally independent of the Bohr set parameters.

3. Probabilistic and Combinatorial Arguments:

   • The construction uses an inductive "triangular array" of blocks.
   • They utilize the Erdős–Turán Lemma and mixing properties of circle rotations to ensure that the approximation sets for different $z \in Z$ are sufficiently independent to achieve full measure (via the Borel–Cantelli lemma in reverse).
   • A key combinatorial hurdle (related to the divergence of reciprocal sums of primes in Bohr sets) is resolved by Manuel Hauke in the appendix, who proves that certain positive-density sets do not contain subsets with bounded GCDs and divergent reciprocal sums, clarifying the limits of the construction.

3. Key Contributions and Results

Theorem 1.1 (Main Result: Unconditional Duffin–Schaeffer Examples)

For any countable sets $Y, Z \subseteq \mathbb{R}$ such that $Z \subseteq \mathbb{R} \setminus \text{span}_{\mathbb{Q}}(\{1\} \cup Y)$, there exists a function $\psi: \mathbb{N} \to \mathbb{R}_{\ge 0}$ such that:

• $\sum \psi(q) = \infty$.
• $\mu(W(\psi, y)) = 0$ for all $y \in Y$.
• $\mu(W(\psi, z)) = 1$ for all $z \in Z$.
• Significance: This removes the conditional assumption from previous work, establishing the existence of such "pathological" functions unconditionally.

Theorem 1.2 (Rogers's Theorem for Real Residues)

For $\epsilon = 2^{-k}$ and any $n \in \mathbb{N}^\ell$, the measure of the real residue system $R(\epsilon, a, n)$ is minimized when $a = 0$:
$$\mu(R(\epsilon, a, n) \cap [0, \text{lcm}(n)]) \ge \mu(R(\epsilon, 0, n) \cap [0, \text{lcm}(n)])$$

• Significance: This extends a classical result on covering systems to the continuous (real) setting, providing the necessary lower bounds for the measure of approximation sets.

Theorem 1.3 (Prime Number Theorem for Bohr Sets)

Let $H^\times$ be the set of points in a Bohr set generated by coprime indices. If a ball $B(v, \epsilon)$ intersects $H^\times$, then the number of primes in the Bohr set $N_v(w, \epsilon)$ up to $X$ satisfies:
$$\#(P \cap N_v(w, \epsilon) \cap [1, X]) \sim \mu_\times(B(v, \epsilon)) \frac{X}{\log X}$$

• Significance: This generalizes the distribution of primes from arithmetic progressions to the more complex geometry of Bohr sets.

Theorem 1.4 (Equidistribution along Bohr-Set Primes)

If $z$ is rationally independent of the Bohr set parameters, the sequence $(pz)_{p \in P \cap N_v(y, \epsilon)}$ is equidistributed modulo 1.

• Significance: This generalizes Vinogradov's theorem, ensuring that primes in these specific sets behave "randomly" enough for the probabilistic arguments in the main proof to hold.

Appendix A (Manuel Hauke)

• Result: Answers a combinatorial question regarding subsets of positive density. It proves that a set $A$ contains a subset $B$ with bounded pairwise GCDs and divergent reciprocal sums if and only if $A$ contains a scaled set of primes $n_0 P$ where $\sum_{p \in P} 1/p = \infty$.
• Significance: This clarifies the structural requirements for the divergence of reciprocal sums in the context of the main paper's construction.

4. Significance and Impact

1. Resolution of a Conditional Problem: The paper provides the first unconditional proof that the inhomogeneous Duffin–Schaeffer phenomenon (zero measure for some parameters, full measure for others, despite divergent sum) holds for arbitrary countable sets of parameters.
2. New Analytic Tools: The development of Real Residue Systems and the extension of Siegel–Walfisz and Vinogradov theorems to Bohr sets creates a new toolkit for studying Diophantine approximation in higher dimensions and non-standard settings.
3. Interplay of Fields: The work deeply integrates Metric Number Theory, Analytic Number Theory (distribution of primes), Ergodic Theory (equidistribution), and Additive Combinatorics (covering systems and GCD properties).
4. Counterexamples: It reinforces the necessity of monotonicity conditions in Khintchine-type theorems, showing that without them, the measure of the approximation set can be arbitrarily manipulated (0 or 1) regardless of the divergence of the sum.

In summary, this paper represents a significant advancement in metric Diophantine approximation, solving a long-standing open problem by bridging the gap between classical covering systems and the distribution of primes in complex geometric sets (Bohr sets).