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Inhomogeneous Approximation by Sums of Roots

This paper establishes that for any real $\beta$ and $\epsilon>0$ , sums of $k$ $d$ -th roots of integers up to $N$ can approximate $\beta$ with an error bound of $O(N^{-k/d+\epsilon})$ , significantly improving upon previous exponents by combining Schmidt's Subspace Theorem with an inhomogeneous transference argument, while also providing explicit constructions for the square-root case.

Original authors: Samuel Korsky

Published 2026-05-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Korsky

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 trying to hit a moving target on a wall with a dart, but you have a very strange set of rules.

The Game
You have a target number, let's call it $\beta$ (beta). It could be any number on the number line, like 3.14 or 100.5. Your goal is to get as close as possible to this target using a specific tool: sums of roots.

You are allowed to pick $k$ numbers (let's call them $b_1, b_2, \dots, b_k$ ). These numbers must be whole numbers (integers) between 1 and some large limit $N$ . You then take the $d$ -th root of each of these numbers (like a square root if $d=2$ , or a cube root if $d=3$ ) and add them all up.

The question is: How close can you get to your target $\beta$ ?

The Old Way vs. The New Way
Before this paper, mathematicians (specifically a researcher named Iyer) had a method to find these numbers. Their method worked well, but it was like trying to hit a bullseye with a slingshot that had a slightly loose rubber band. They could get close, but the "closeness" (the error) dropped off relatively slowly as you increased your limit $N$ .

In this paper, Samuel Korsky introduces a new, sharper slingshot. He proves that you can get much closer to the target than previously thought possible.

The Magic Trick: The "Subspace" and the "Transfer"
Korsky's proof uses two main ideas, which he combines like a two-step magic trick:

The "No-Go" Zone (Schmidt's Subspace Theorem):
Imagine you have a group of distinct prime numbers (like 2, 3, 5, 7...). If you take their roots and multiply them by a whole number, they usually land in very specific, "messy" spots on the number line, never perfectly aligning with whole numbers. Korsky uses a famous mathematical theorem (Schmidt's Subspace Theorem) to prove that these roots are stubbornly resistant to lining up perfectly. They create a "No-Go Zone" where they simply refuse to be too close to integers unless you use huge numbers. This establishes a baseline of how "spread out" these numbers are.
The "Transfer" (The Inhomogeneous Argument):
Once he knows how spread out the roots are, he uses a "transfer" argument. Think of this like a translator. He takes the information about how the roots avoid integers (the "No-Go Zone") and translates it into a guarantee that you can find a combination of them that lands very close to any target you want.

It's like knowing that a specific type of bird never lands on a certain branch. Because you know exactly where they don't land, you can predict exactly where they will land if you shake the tree just right.

The Result: A Much Tighter Shot
The paper proves that for any target, you can find your numbers $b_1$ through $b_k$ such that the distance to the target is incredibly small. Specifically, the error shrinks at a rate of roughly $1/N^{k/d}$ .

Why this matters: If you have more roots to add ( $k$ is larger), you can hit the target with much higher precision. If the roots are "higher" (like cube roots instead of square roots), it's slightly harder, but the new formula still beats the old record.
The Catch: The proof is "ineffective." This means Korsky can prove that such numbers exist and that they are very close, but he doesn't give you a specific recipe to find them easily. It's like proving a treasure exists on an island without giving you the map coordinates. You know it's there, but finding it might take a long time.

The "Perfect" Shot (The Conjecture)
Korsky also suggests a "Holy Grail" version of this problem. He guesses that with the right combination of numbers, you could get even closer—specifically, the error could shrink at a rate of $1/N^{k - 1/d}$ .

He can't prove this for all cases yet, but he shows it works for specific, simple scenarios (like square roots with 2, 3, or 4 numbers added together). He does this by carefully constructing numbers that cancel each other out perfectly, like balancing weights on a scale until only a tiny, tiny fraction remains.

In Summary
This paper is a mathematical victory lap. It shows that by using a powerful, high-level theorem about how numbers behave, we can prove that sums of roots can approximate any target number much more accurately than we previously knew. While we can't always easily find the specific numbers to do this (the proof is non-constructive), we now know the theoretical limit of how close we can get is much tighter than before.

Technical Summary: Inhomogeneous Approximation by Sums of Roots

Problem Statement
The paper addresses the problem of inhomogeneous Diophantine approximation involving sums of $d$ -th roots. Specifically, given fixed integers $d \ge 2$ and $k \ge 1$ , a real target $\beta$ , and a large bound $N$ , the goal is to find integers $1 \le b_1, \dots, b_k \le N$ such that the distance from the sum of their $d$ -th roots to $\beta$ is minimized:
$\left\| \sum_{j=1}^k b_j^{1/d} - \beta \right\|$
where $\|x\|$ denotes the distance from $x$ to the nearest integer. The paper seeks to establish the optimal exponent $\gamma$ such that this distance is bounded by $O(N^{-\gamma})$ .

Prior work by Iyer (2025) established an effective inhomogeneous exponent of at least $(k-d+1)/d^2$ and a sharper "full-basis" exponent $\gamma_{k,d} = d^{\lfloor \log_d(k+1) \rfloor - 1}/d$ . The latter achieves the value $k/d$ only when $k+1$ is a power of $d$ . The paper aims to improve these exponents for general $k$ and $d$ , particularly in cases where $k+1$ is not a power of $d$ , albeit at the cost of effectivity.

Methodology
The proof of the main theorem relies on two primary components:

Dual Lower Bound via Schmidt's Subspace Theorem:
The authors utilize Schmidt's Subspace Theorem to establish a dual lower bound for linear forms involving distinct prime radicals. By selecting distinct primes $p_1, \dots, p_k$ and setting $\theta_i = p_i^{1/d}$ , they invoke the linear independence of $1, \theta_1, \dots, \theta_k$ over $\mathbb{Q}$ (a result due to Besicovitch). This yields a lower bound of the form:
$\max_{1 \le i \le k} \| h \theta_i \| \gg_{d,k,\eta} |h|^{-1/k - \eta}$
for any $\eta > 0$ and non-zero integer $h$ . This step is non-effective, as it relies on the Subspace Theorem.
Inhomogeneous Transference:
The authors employ a covering transference argument (Lemma 3.2) to convert the dual lower bound into an inhomogeneous approximation result. Using the geometry of numbers (specifically Mahler's transference principle and the covering radius of a lattice), they show that if the dual bound holds with exponent $\sigma$ , then for any target $\beta$ and parameter $Q$ , there exist integers $q_1, \dots, q_k$ with $|q_i| \le Q$ such that:
$\left\| \sum_{i=1}^k q_i \theta_i - \beta \right\| \ll Q^{-1/\sigma}$
By shifting the coefficients to be positive and redefining the radicands as $b_i = p_i c_i^d$ (where $c_i$ are derived from the $q_i$ ), the problem is transformed back into a sum of $d$ -th roots with radicands bounded by $N$ .

Key Results

Theorem 1.1 (Main Theorem): For any fixed $d \ge 2$ , $k \ge 1$ , and $\varepsilon > 0$ , there exists a constant $C_{d,k,\varepsilon}$ such that for every $N \ge 2$ and $\beta \in \mathbb{R}$ , there exist integers $1 \le b_1, \dots, b_k \le N$ satisfying:
$\left\| \sum_{j=1}^k b_j^{1/d} - \beta \right\| \le C_{d,k,\varepsilon} N^{-k/d + \varepsilon}$
This result provides an exponent of $k/d - \varepsilon$ . It improves upon Iyer's exponent $(k-d+1)/d^2$ and the full-basis exponent $\gamma_{k,d}$ whenever $k+1$ is not a power of $d$ . In the specific case where $k+1$ is a power of $d$ , Iyer's effective construction already achieves $k/d$ , whereas this theorem loses an arbitrary $\varepsilon$ and is ineffective.
Conjecture 1.2 (Uniform Exponent): The authors propose a stronger conjecture that the optimal uniform exponent is $k - 1/d$ . This suggests that the current result of $k/d$ is not the final limit, but rather a consequence of the specific algebraic structure (sparse family of radicands) used in the proof. The conjecture posits:
$\sup_{\beta \in \mathbb{R}/\mathbb{Z}} \min_{1 \le b_1, \dots, b_k \le N} \left\| \sum_{j=1}^k b_j^{1/d} - \beta \right\| \ll_{d,k,\varepsilon} N^{-(k-1/d) + \varepsilon}$
Explicit Constructions for $d=2$ : In Section 5, the paper provides explicit integer-target constructions for the square-root case ( $d=2$ ) with $k=2, 3, 4$ . Using Taylor expansions and coefficient cancellation, the authors construct sums of square roots that approximate an integer with an error of order $M^{-(2k-1)}$ , where the radicands are of size $M^2$ . This yields an exponent of $N^{-(k-1/2)}$ , matching the conjectural value for these specific cases.

Significance and Claims
The paper claims to provide a different proof for inhomogeneous exponents in sums of roots, shifting the methodology from Iyer's "full radical basis" approach to a general algebraic Diophantine approximation framework using Schmidt's Subspace Theorem.

The primary significance lies in the improvement of the exponent for cases where $k+1$ is not a power of $d$ . The authors note that while their result is ineffective (due to the Subspace Theorem), it achieves a better exponent than previous effective constructions in those specific regimes. The paper also highlights the gap between the proven exponent ( $k/d$ ) and the conjectured optimal exponent ( $k-1/d$ ), suggesting that the latter requires exploiting more "local freedom" in the choice of radicands beyond the sparse family used in the main proof. The explicit constructions for $d=2$ serve as evidence supporting the conjectured uniform exponent.

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