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Integer parts of real powers in two Erd\H{o}s problems of Romanoff type

This paper resolves two Erdős-type additive problems by proving a quantitative metric lower bound for the density of integers representable as a prime plus an integer part of a real power, and establishing a density-one analogue for square-free integers plus a real power, while also demonstrating that positive lower density of exceptions persists for specific bases like the golden ratio.

Original authors: Yuchen Ding

Published 2026-07-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuchen Ding

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

Imagine you are a detective trying to solve a massive puzzle involving numbers. In the world of mathematics, there is a famous game called "additive number theory," where the goal is to see if you can build every number by adding two specific types of ingredients together. One ingredient is usually a "prime number" (numbers like 2, 3, 5, 7 that can only be divided by 1 and themselves) or a "square-free number" (numbers that aren't divisible by any perfect square like 4, 9, or 16). The other ingredient is a "sparse sequence," which is a list of numbers that gets very thin, very fast, like powers of 2 (2, 4, 8, 16...).

For decades, mathematicians have been asking: If you take a prime number and add a number from this thin list, can you make every large number? Or, at the very least, can you make almost every number? This is like asking if you can build a house using a specific type of brick and a specific type of wood. Sometimes the answer is "yes, you can build almost any house," and sometimes the answer is "no, there are certain house shapes you just can't build." The paper you are about to read dives into this question, but with a twist: instead of using whole numbers for the thin list, the author uses the "integer parts" of powers of real numbers (like 1.5, 2.7, or the golden ratio). It's like asking if you can build houses using bricks that are cut from a continuous stream of wood rather than pre-cut planks.

The Story of Two Puzzles

This paper, written by Yuchen Ding, tackles two different versions of this "building blocks" puzzle. The author is trying to figure out if we can cover the entire number line (or at least most of it) by adding primes or square-free numbers to these special "real power" sequences.

Puzzle 1: The Prime Numbers and the "Almost Always" Rule

The first problem asks: If you pick a random number greater than 1 (let's call it $y$ ) and create a list of numbers by taking the whole number part of $y^1, y^2, y^3$ , and so on, can you add these to prime numbers to get almost every other number?

The author proves a very strong result here: For almost every number you pick, the answer is a resounding "yes." If you pick a number $y$ at random, it is mathematically guaranteed that the set of numbers you can build (Prime + Integer Part of $y^k$ ) will have a positive lower asymptotic density. In plain English, this means you won't just be able to build a few houses; you'll be able to build a massive, infinite neighborhood of them, ensuring that no matter how far out you look, you will always find a significant chunk of numbers you can construct. The paper even gives a precise formula for the minimum size of this chunk, showing that the more "spread out" your powers are (the larger $y$ is), the fewer numbers you can build, but you will always build a significant portion.

However, the paper also reveals a fascinating "exception." Just because it works for almost every number doesn't mean it works for every number. The author constructs a specific, tricky example using the Golden Ratio (that famous number $\phi \approx 1.618$ often found in nature and art). When you use the Golden Ratio, the "integer parts" of its powers behave in a very special, rhythmic way (they are closely related to Lucas numbers). The author proves that if you use this specific number, there is a positive proportion of integers that simply cannot be built. It's like finding a specific type of wood that, no matter how you try, leaves a gap in your wall that you can never fill. This is a concrete proof that the "Golden Ratio" is a "bad actor" in this game, leaving a permanent hole in the coverage.

Puzzle 2: The Square-Free Numbers and the "Magic Number"

The second problem swaps primes for "square-free numbers" (numbers not divisible by 4, 9, 16, etc.). A famous mathematician named Erdős once guessed that if you add powers of 2 to square-free numbers, you could build every large odd number. This specific version of the puzzle is still unsolved and considered very hard.

Instead of solving the hard version with powers of 2, the author asks a slightly different question: Can we find some real number $a$ (between 2 and 3) such that if we use its powers, we can build almost every positive integer?

The answer is yes. The author proves that there exists at least one "magic number" $a$ in that range. By carefully choosing this number, the author shows that the "integer parts" of its powers can be forced to land in just the right spots to fill in the gaps left by the square-free numbers. It's like finding a secret key that, when turned, unlocks the ability to build nearly every house in the city. The paper doesn't tell us exactly what this number is (it proves it exists but doesn't write it down as a simple decimal), but it proves that such a number is out there, waiting to be found.

The Big Picture

In summary, this paper is a tour de force of mathematical detective work. It shows that:

Generally, it works: If you pick a random base for your powers, you can almost always build a huge collection of numbers by adding them to primes, guaranteeing a positive lower density.
But watch out for the Golden Ratio: There is one specific, famous number where the pattern breaks, leaving a permanent gap.
A magic solution exists: For the square-free problem, while we can't solve the original "powers of 2" version yet, we know there is a "magic" real number that solves a very similar, density-one version of the problem.

The author uses a mix of "metric" arguments (looking at what happens on average for random numbers) and "constructive" arguments (building specific examples to prove gaps or solutions). The results are not just guesses or computer simulations; they are rigorous mathematical proofs. The paper confirms that while the universe of numbers is vast and sometimes tricky, there are deep, underlying rules that allow us to cover almost everything, provided we choose our ingredients just right.

Technical Summary: Integer Parts of Real Powers in Two Erdős Problems of Romanoff Type

Overview and Problem Statement
This paper investigates two additive problems within the framework of the Erdős–Romanoff line of inquiry. The central theme is the replacement of rigid sequences of integral powers ( $a^k$ ) with the integer parts of powers of a real base ( $\lfloor y^k \rfloor$ ). This modification preserves the logarithmic sparseness of the summand while introducing a real parameter that allows for both metric (probabilistic) and constructive approaches. The paper addresses two specific questions:

The Prime Problem (Erdős–Kalmár): Does the set $S_y = \{p + \lfloor y^k \rfloor : p \in \mathbb{P}, k \ge 1\}$ have positive lower asymptotic density for every real $y > 1$ ?
The Square-Free Problem: Can every sufficiently large integer be represented as the sum of a square-free integer and a power of two ( $n = q + 2^k$ )? While the fixed-base version remains open, the paper seeks a variable-base analogue where $2^k$ is replaced by $\lfloor a^m \rfloor$ .

Methodology
The paper employs three distinct methodological approaches corresponding to its three main theorems:

Metric Number Theory and Second-Moment Analysis (Theorem 1.1):
To establish positive density for almost all real bases, the author combines a deterministic Romanoff second-moment criterion with metric estimates.
- Second-Moment Criterion: A quantitative lemma (Lemma 3.1) is derived, linking the lower density of $P+A$ to the counting function of $A$ and the weighted pair correlations of differences in $A$ . The weight $W(h) = \prod_{p|h}(1+1/p)$ accounts for the distribution of primes in arithmetic progressions via the Selberg sieve.
- Metric Control: The core technical contribution is proving that for almost all $y$ , the sequence $\lfloor y^k \rfloor$ satisfies the necessary pair-correlation estimates. This involves analyzing the distribution of $\lfloor y^k \rfloor \pmod d$ . The proof utilizes the Erdős–Turán inequality and estimates exponential sums $\sum e(h y^k / d)$ . By separating divisors into "small" (handled by equidistribution arguments) and "large" (handled by derivative estimates of $y^\ell - y^k$ ), the author shows that the weighted pair correlations match the random model expectation $\zeta(2)/2\zeta(4)$ for almost all $y$ .
Covering Congruences and Lucas Numbers (Theorem 1.2):
To demonstrate that density-one coverage is not guaranteed even in the real-base setting, the paper constructs an "exceptional base."
- Golden Ratio Connection: The author exploits the identity between $\lfloor \phi^k \rfloor$ (where $\phi$ is the golden ratio) and Lucas numbers ( $L_k$ ). Specifically, $\lfloor \phi^k \rfloor$ equals $L_k$ or $L_k - 1$ depending on the parity of $k$ .
- Finite Covering: A finite set of congruences is constructed to cover all exponents $k$ . By selecting a specific arithmetic progression for the target integers $n$ , the author ensures that for any representation $n = p + \lfloor \phi^k \rfloor$ , the prime $p$ must be divisible by one of a finite set of small primes. This forces $p$ to be one of those small primes, limiting the number of representable integers in the progression to $O(\log x)$ , thereby proving a positive proportion of integers are missed.
Nested-Interval Construction and Residue Covering (Theorem 1.3):
For the square-free problem, the paper proves the existence of a real base $a \in (2,3)$ such that $\lfloor a^m \rfloor$ acts as an additive complement to the square-free integers with density one.
- Residue Covering: The proof constructs a finite set of shifts modulo $Q_z = \prod_{p \le z} p^2$ that intersect the set of square-free residues $R_z$ with high multiplicity.
- Nested Intervals: A constructive argument (Lemma 5.2) is used to realize a countable sequence of prescribed congruence classes as values of $\lfloor a^m \rfloor$ for a single real number $a$ . This is achieved by defining a nested sequence of closed intervals $I_0 \supset I_1 \supset \dots$ where each step fixes a specific exponent $m$ and value $t = \lfloor a^m \rfloor$ to satisfy a required congruence.
- Density Argument: The construction ensures that for large $x$ , the "exceptional" integers (those not representable) are dominated by those failing to be square-free modulo large primes, a set of density zero.

Key Results

Theorem 1.1 (Metric Prime Theorem): For Lebesgue almost all $y > 1$ , the set $S_y = \{p + \lfloor y^k \rfloor\}$ has a lower asymptotic density $\delta_y$ satisfying:
$\delta_y \ge \frac{1}{\log y + C_{sv} \zeta(2)/\zeta(4)}$
where $C_{sv}$ is the absolute Selberg-sieve constant. The order $1/\log y$ is shown to be optimal as $y \to \infty$ .
Theorem 1.2 (Exceptional Base for Primes): For the specific base $y = \phi = (1+\sqrt{5})/2$ , the set of integers not representable as $p + \lfloor \phi^k \rfloor$ has positive lower density. Specifically:
$\#\{n \le x : n \notin P + \{\lfloor \phi^k \rfloor\}\} \ge \frac{x}{7845692610} - O(\log x)$
Theorem 1.3 (Square-Free Density-One Complement): There exists a real number $a \in (2, 3)$ such that the set of integers not representable as $q + \lfloor a^m \rfloor$ (where $q$ is square-free) has density zero:
$\#\{n \le x : n \notin Q + \{\lfloor a^m \rfloor\}\} = o(x)$
This result is extended in Corollary 5.3 to $r$ -free integers for any $r \ge 2$ .

Significance and Claims
The paper claims to resolve the Erdős–Kalmár conjecture for almost all real bases, providing a precise quantitative lower bound that depends on the base $y$ . It clarifies the boundary of the Romanoff phenomenon by showing that while positive density is generic, full coverage (density one) is not guaranteed for all real bases, as demonstrated by the golden ratio counterexample.

Regarding the square-free problem, the paper does not solve the fixed-base conjecture ( $2^k + q$ ) but provides a "density-one analogue" in the variable-base setting. The significance lies in demonstrating that the flexibility of a real parameter allows one to impose congruence conditions unavailable to fixed bases, thereby overcoming the obstructions that make the fixed-base problem difficult. The work unifies metric number theory, sieve methods, and constructive covering arguments to address long-standing additive problems involving sparse sequences.

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