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Equiangular lines via improved eigenvalue multiplicity
Combinatorics · Submitted 24 Sep 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the answer equals $r-1+\left\lfloor\frac{r-1}{k-1}\right\rfloor,$ provided that $r$ is at least exponential in a polynomial in $k$.
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The abstract, sentence by sentence
A family of lines passing through the origin in an inner product space is said to be equiangular if every pair of lines defines the same angle. In 1973, Lemmens and Seidel raised what has since become a central question in the study of equiangular lines in Euclidean spaces. They asked for the maximum number of equiangular lines in $\mathbb{R}^r$ with a common angle of $\arccos{\frac{1}{2k-1}}$ for any integer $k \geq 2$. We show that the answer equals $r-1+\left\lfloor\frac{r-1}{k-1}\right\rfloor,$ provided that $r$ is at least exponential in a polynomial in $k$. This improves upon a recent breakthrough of Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. (2) 194 (2021), no. 3, 729-743], who showed that this holds for $r$ at least doubly exponential in a polynomial in $k$. We also show that for any common angle $\arccos{\alpha}$, the answer equals $r+o(r)$ already when $r$ is superpolynomial in $1/\alpha \to \infty$. The key new ingredient underlying our results is an improved upper bound on the multiplicity of the second-largest eigenvalue of a graph. In one of the regimes, this improves and significantly extends a result of McKenzie, Rasmussen, and Srivastava [STOC 2021, pp. 396-407].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A family of lines passing through the origin in an inner product space is said to be equiangular if every pair of lines defines the same angle. | background 1.00 | 0.04 | 0.01 | asserted 0.99 |
| s1 | In 1973, Lemmens and Seidel raised what has since become a central question in the study of equiangular lines in Euclidean spaces. | background 1.00 | 0.02 | 0.01 | asserted 0.98 |
| s2 | They asked for the maximum number of equiangular lines in $\mathbb{R}^r$ with a common angle of $\arccos{\frac{1}{2k-1}}$ for any integer $k \geq 2$. | background 0.99 | 0.03 | 0.04 | asserted 0.78 |
| s3 | We show that the answer equals $r-1+\left\lfloor\frac{r-1}{k-1}\right\rfloor,$ provided that $r$ is at least exponential in a polynomial in $k$. | result 0.99 | 0.96 | 0.86 | asserted 1.00 |
| s4 | This improves upon a recent breakthrough of Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. | contribution 0.71 | 0.74 | 0.05 | asserted 1.00 |
| s5 | (2) 194 (2021), no. 3, 729-743], who showed that this holds for $r$ at least doubly exponential in a polynomial in $k$. | background 0.55 | 0.04 | 0.05 | asserted 0.99 |
| s6 | We also show that for any common angle $\arccos{\alpha}$, the answer equals $r+o(r)$ already when $r$ is superpolynomial in $1/\alpha \to \infty$. | result 0.99 | 0.96 | 0.16 | asserted 1.00 |
| s7 | The key new ingredient underlying our results is an improved upper bound on the multiplicity of the second-largest eigenvalue of a graph. | method 0.65 | 0.91 | 0.06 | asserted 1.00 |
| s8 | In one of the regimes, this improves and significantly extends a result of McKenzie, Rasmussen, and Srivastava [STOC 2021, pp. 396-407]. | contribution 0.60 | 0.84 | 0.03 | asserted 1.00 |
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The paper
Original abstract
A family of lines passing through the origin in an inner product space is said to be equiangular if every pair of lines defines the same angle. In 1973, Lemmens and Seidel raised what has since become a central question in the study of equiangular lines in Euclidean spaces. They asked for the maximum number of equiangular lines in $\mathbb{R}^r$ with a common angle of $\arccos{\frac{1}{2k-1}}$ for any integer $k \geq 2$. We show that the answer equals $r-1+\left\lfloor\frac{r-1}{k-1}\right\rfloor,$ provided that $r$ is at least exponential in a polynomial in $k$. This improves upon a recent breakthrough of Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. (2) 194 (2021), no. 3, 729-743], who showed that this holds for $r$ at least doubly exponential in a polynomial in $k$. We also show that for any common angle $\arccos{\alpha}$, the answer equals $r+o(r)$ already when $r$ is superpolynomial in $1/\alpha \to \infty$. The key new ingredient underlying our results is an improved upper bound on the multiplicity of the second-largest eigenvalue of a graph. In one of the regimes, this improves and significantly extends a result of McKenzie, Rasmussen, and Srivastava [STOC 2021, pp. 396-407].
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What it cites · 5 sentences without a confident label
Builds on 1 paper
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In [ref] , the second author extended the bound of Yu to all $r ≤r_α ≈1/4α^4$ and showed that the linear bound of [ref] holds for all $r≥r_α$, thereby extending it from the exponential to the polynomial regime.
2110.15842 · Equiangular lines · 0.93
Compares with 1 paper
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This improves upon the previously best-known doubly exponential constraint on $r$ from [ref] .
2110.15842 · Equiangular lines · 0.99
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