Sums of square roots that are close to an integer

Abstract: Let $k \in \mathbb{N}$ and suppose we are given $k$ integers $1 \leq a_1, \dots, a_k \leq n$. If $\sqrt{a_1} + \dots + \sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\gtrsim n^{-1/2}$. Angluin-Eisenstat observed the bound $\gtrsim n^{-3/2}$ when $k=2$. We prove there is a universal $c>0$ such that, for all $k \geq 2$, there exists a $c_k > 0$ and $k$ integers in $\left{1,2,\dots, n\right}$ with $$ 0 <|\sqrt{a_1} + \dots + \sqrt{a_k} | \leq c_k\cdot n^{-c \cdot k^{1/3}},$$ where $| \cdot |$ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: even for $k=3$, constructing explicit examples of integers whose square root sum is nearly an integer appears to be nontrivial.

• The paper establishes a universal bound, proving there exists a constant c > 0 such that for any k ≥ 2, one can choose integers whose square root sum is within c · k^(1/3) · n^(-c · k^(1/3)) of an integer.
• It reveals that for k ≥ 3 the construction of such integer selections becomes nontrivial, marking a significant increase in complexity compared to simpler cases.
• By employing Vinogradov's method and van der Corput estimates, the study provides non-trivial lower bounds that advance numerical approximation and algorithmic efficiency.

Analyzing Close Approximations of Sums of Square Roots to Integers

The paper by Stefan Steinerberger explores the problem of determining how closely the sum of the square roots of integers can approach an integer. This study builds on the foundational investigation into square-root sums, a well-explored topic in numerical analysis and computational complexity theory, and offers new insights particularly when the number of summed terms, $k$, exceeds two. The problem arises in various computational contexts, for instance in calculating whether one complex arithmetic expression exceeds another in value.

Key Results

1. Universal Bound: The paper establishes that there is a universal constant $c > 0$, such that for any natural number $k \geq 2$, there exists a choice of integers among $\{1, 2, ..., n\}$ such that the sum of their square roots is close to an integer by a factor of $c \cdot k^{1/3} \cdot n^{-c \cdot k^{1/3}}$.
2. Complexity for $k \geq 3$: The research highlights that when $k = 3$, it becomes nontrivial to explicitly construct integers whose square root sums closely approach integers. This discovery breaks with simpler approximations available when $k = 1$ or $k = 2$, demonstrating the increased complexity as the number of terms increases.
3. Lower Bounds: Using methods like the Vinogradov method and van der Corput estimates, the author provides non-trivial lower bounds on the distance of these sums from integers, noting that they extend beyond simple squaring techniques utilized for smaller $k$.
4. Combinatorial Approach: The research engages combinatorial constructs, also reflecting on existing number theory frameworks such as the Prouhet–Tarry–Escott problem, contributing to understanding the configurations that lead these sums close to integers.

Implications and Future Directions

The implications of this work are both practical and theoretical. On a theoretical level, it advances our understanding of the structural behavior of irrational sums in modular arithmetic, contributing to both number theory and computational complexity. Practically, insights from this work can enhance our approaches to numerical computation and algorithmic efficiency by offering refined methods for approximating irrational numbers.

Future exploration might delve further into improving the provided bounds or extending these results to broader classes of functions beyond those involving integer square roots. Additionally, investigating alternative methods of constructing square root sums that approach integers could yield new pathways for both applied computational tasks and theoretical inquiries in discrete mathematics.

This work opens avenues for continued examination of the gap structure within sets of irrational sums, which may have exceptional adjusted statistical properties, compared to predicted typical behavior, suggesting richer underlying algebraic or combinatorial principles yet to be discovered.