Bohr Set Visualizer

Description

The Bohr Set Visualizer is a Streamlit research app plus Python numerics for Bohr sets $\Lambda_{\Theta,\varepsilon}$ inside $\mathbb{Z}_{N}$, motivated by the corners theorem proof for general abelian groups in Jaber–Liu–Lovett–Ostuni–Sawhney (arXiv:2504.07006). It turns dense definitions (regular Bohr sets, relative sifting, $(d,\eta)$ chains, $\ell_{1}$-spreadness, dual Fourier spectrum, and qualitative links to Sec. 7 and Sec. 8) into plots and sliders you can change live.

Scope. The implementation fixes $G = \mathbb{Z}_{N}$ with explicit characters $\chi_{\theta}(x) = e^{2\pi i, \theta x / N}$; the README also summarizes how such abelian inputs relate to extensions beyond abelian groups and coloring results stated in the paper.

GitHub “About” (one line, copy-paste):
Interactive Streamlit tool: Bohr sets in ℤ_N, regularity, sub-Bohr sifting, increment chains, ℓ₁-spread and Fourier views, pair-health proxy, and coloring pedagogy—aligned with arXiv:2504.07006 (quasipolynomial corners).

Math in this file uses GitHub’s $…$ / $$…$$ syntax so formulas render in the GitHub web UI. Subscripts are braced in LaTeX and escaped for Markdown (e.g. $\mathbb{Z}\_{N}$, $\chi\_{\theta}$, $\ell\_{1}$): each subscript _ is written as \_ in the .md source so CommonMark does not strip _ before KaTeX runs.

Streamlit UI (screenshot)

Live controls for $N$, the ball norm (max / mean), frequency set $\Theta$, nested Bohr containers, and radius $\varepsilon$; tabs below (not fully visible here) host regularity, random walk, sifting, chains, $\ell_{1}$-spread, Fourier, and pair-health views. Example state: $N=180$, $d=2$, arithmetic progression $\Theta={7,18}$, $\varepsilon=0.08$.

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Paper reference (PDF)

Copy	Path
Current (Cursor workspace)	c:\Users\adith\AppData\Roaming\Cursor\User\workspaceStorage\e0f943467b0a7cf1dc5dcf6b6f4194bc\pdfs\b0e9945c-06fe-4c99-89b3-c9e09200ecec\2504.07006v2 (1).pdf
Other workspace copies	…\0b984680-ef1a-48bb-bf61-1afb12277129\…, …\ba6baa42-86cb-4ab0-8c4b-3d1c72ba8949\…, …\b6b3e01f-9e8f-4de2-bc0a-3587d1e9a389\…, …\32e81826-9a93-4e51-9a32-2e7cd4f2312a\…

All refer to the same arXiv revision: 2504.07006v2 (abstract).

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Section 6 structure (paper)

The corners proof for general abelian groups replaces subspaces with Bohr sets; Section 6 develops Bohr sets, spreadness, and pseudorandomization. Dependency outline:

flowchart TB
  S6["Sec. 6 — Bohr sets, algebraic spreadness, pseudorandomization"]
  S6 --> A["6.1 Bohr sets"]
  A --> D61["Def 6.1 Λ_{Θ,ε}, dilation cΛ"]
  A --> L62["Lemma 6.2 mass bound"]
  A --> D63["Def 6.3 regular Bohr sets"]
  A --> L64["Lemma 6.4 ∃α Λ regular"]
  A --> L65["Lemma 6.5 shift invariance"]
  A --> D66["Def 6.6 (d,η)-small / exact sequences"]
  S6 --> B["Later in Sec. 6: grid norms, spreadness, …"]
  B --> D613["Def 6.13 ℓ₁-spreadness"]
  B --> D615["Def 6.15 algebraic spreadness"]

Loading

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Non-abelian groups (Section 1.1): where $\mathbb{Z}_{N}$ fits

The main theorem is stated for finite abelian $G$, but the introduction explains a striking extension: quasipolynomial corner bounds for abelian groups imply analogous bounds for every finite group $G$, including non-abelian ones. The pipeline (Fox-type argument, as discussed in Sec. 1.1) is:

1. Find a large abelian subgroup $H \leq G$ (every large finite group contains a proportional abelian piece).
2. Run the abelian theory on $H$ — Bohr sets, relative sifting, grid norms, and the diagonal “$\ell_{1}$-spread” phenomena you simulate here are native to that step.
3. Average / lift from $H$ back to $G$ so that dense corner-free structure in $G$ cannot evade the abelian obstruction.

Implementation link. This repository only draws $G = \mathbb{Z}_{N}$, but that model is exactly the kind of abelian building block that feeds the general proof: Bohr sets are the right containers inside $H$; the lift to arbitrary $G$ is conceptual, not a different codebase.

Schematic (Sec. 1.1). The paper explains how abelian bounds transfer to all finite $G$ via a large abelian $H$ and an averaging / lifting step. This figure is a storytelling aid, not a literal algorithm.

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Coloring and the “tower” warning (Section 8)

The paper’s coloring corollary (Corollary 1.2, Sec. 8) addresses $G \times G \times G$ with about $L \asymp \log\log\log \lvert G\rvert$ colors, forcing a monochromatic corner. The authors stress that with only doubly-logarithmic density savings (as in Shkredov’s classical regime), one would face tower-type losses in how many colors can be handled — i.e. the quantitative strength of the new bound is what makes a polylogarithmic-in-log color count possible.

Visualizer add-on. Under Tab 1 → “Coloring strip”, the app assigns an $L$-coloring of $\mathbb{Z}_{N}$ (either $x \bmod L$ or i.i.d. random) and reports how many colors appear on the current Bohr set $\Lambda$, whether $\Lambda$ is monochromatic, and the exact probability $L^{1-\lvert \Lambda\rvert}$ that a fully random coloring makes $\Lambda$ monochromatic. That toy is 1-dimensional; it is meant only to build intuition for why density decay stronger than $(\log\log \lvert G\rvert)^{-c}$ matters for multicolor statements.

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Quantitative arc: why 2025 is a breakthrough

Aspect	Shkredov (2006) corner bound	Jaber et al. (2025)
Strength	Doubly logarithmic: density $\ll (\log\log \lvert G\rvert)^{-c}$	Quasipolynomial: $\lvert A\rvert \le \lvert G\rvert^2 \exp\bigl(-(\log \lvert G\rvert)^{\Omega(1)}\bigr)$
Core mechanism	$L^2$ / energy-style increment	Relative sifting, Bohr containers, Gowers grid norms, asymmetric treatment of the diagonal $D$ ($\ell_{1}$-spread)
Diagonal	(Symmetric finite-field intuition)	“Diagonal drop”: $D$ is not assumed algebraically spread like $X,Y$

The plot below is illustrative only (exponents are not those in the theorems). It compares the shape of a doubly-logarithmic saving with a quasipolynomial-type saving in $\lvert G\rvert$.

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How this repo sits in a larger “suite” (conceptual map)

Focus	Role in the story
Behrend-type constructions	Lower bounds — how dense a corner-free set can still be.
Grid norms / Von Neumann lemmas	Triggers — when lack of uniformity forces structured increments.
Bohr sets (this repo)	Containers — soft structured neighborhoods inside general abelian groups.
Exactly-$N$ / communication	Applications — complexity consequences stated in the paper’s abstract.

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Definitions from the paper (Section 6.1)

Definition 6.1 (Bohr set). Let $\varepsilon \in \mathbb{R}^+$, let $G$ be a finite abelian group, and $\Theta = (\Theta_1,\ldots,\Theta_d)$ with $\Theta_i \in \widehat{G}$ homomorphisms $G \to \mathbb{R}/\mathbb{Z}$. The Bohr set is

$$ \Lambda = \Lambda_{\Theta,\varepsilon} = \bigcap_{i=1}^{d} \Big{ x \in G : \bigl\lVert \Theta_i(x) \bigr\rVert_{\mathbb{R}/\mathbb{Z}} \le \varepsilon \Big}, $$

where $\lVert x \rVert_{\mathbb{R}/\mathbb{Z}} = \min_{z \in \mathbb{Z}} \lvert x - z \rvert$. For $c > 0$, dilation is $c\Lambda_{\Theta,\varepsilon} = \Lambda_{\Theta,,c\varepsilon}$. The paper calls $d$ the dimension and $\varepsilon$ the radius $\nu(\Lambda)$.

Lemma 6.2. If $\Lambda = \Lambda_{\Theta,\varepsilon}$ has dimension $d$ and radius $\varepsilon$, then

$$ \lvert \Lambda \rvert \ge \varepsilon^d \lvert G \rvert. $$

Definition 6.3 (Regular). A Bohr set $\Lambda = \Lambda_{\Theta,\varepsilon}$ of dimension $d$ is regular if for all $\lvert c \rvert \le 1/(100d)$,

$$ 1 - 100d\lvert c \rvert ;\le; \frac{\bigl\lvert (1+c)\Lambda \bigr\rvert}{\lvert \Lambda \rvert} ;\le; 1 + 100d \lvert c \rvert. $$

Lemma 6.5. Let $f$ be $1$-bounded and $\Lambda$ a regular Bohr set of dimension $d$. If $\lvert c \rvert \le 1/(100d)$ and $n' \in c\Lambda$, then

$$ \mathbb{E}_{n \sim \Lambda}, f(n) ;=; \mathbb{E}_{n \sim \Lambda}, f(n + n') ;+; \mathcal{O}(c d). $$

This captures approximate shift-invariance when the shift lies in a smaller dilate $c\Lambda$.

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Cyclic group model ($G = \mathbb{Z}_{N}$)

Characters are $\chi_{\theta}(x) = e^{2\pi i, \theta x / N}$ with $\theta \in {0,\ldots,N-1}$. This repo implements

$\ell_{\infty}$ (max) ball:

$$ \max_{\theta \in \Theta}; \Bigl\lVert \frac{\theta x}{N} \Bigr\rVert_{\mathbb{R}/\mathbb{Z}} < \varepsilon . $$

Mean / $\ell_{1}$ ball:

$$ \frac{1}{\lvert \Theta \rvert} \sum_{\theta \in \Theta} \Bigl\lVert \frac{\theta x}{N} \Bigr\rVert_{\mathbb{R}/\mathbb{Z}} < \varepsilon . $$

These match Definition 6.1 once each $\theta x / N$ is interpreted in $\mathbb{R}/\mathbb{Z}$ (see bohr_set.py).

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Research simulation: mapping to the paper

The Streamlit app is not only drawing Bohr sets; it simulates mechanisms from the proof: relative sifting inside a sparse container, nested Bohr “zoom” chains, and Definition 6.13 $\ell_{1}$-spreadness.

Sub-Bohr sifting and relative density (Sections 2.2, 3.5)

Problem (informal). Classical sifting bounds degrade with the density of a majorant in all of $G$. If $A$ is tiny in $G$, crude estimates lose useful factors.

What the tool reports. For an outer Bohr container $B_1$ and sparse $A \subset B_1$, compare global density $\lvert A\rvert / \lvert G\rvert$ to relative density $\lvert A\rvert / \lvert B_1\rvert$, and track how much of $A$ falls into a smaller inner Bohr neighborhood $B_2$. That is the right mental model for relative sifting: locating a sub-instance where a set that is sparse globally is still dense enough relative to $B_1$ (and refinable inside $B_2$) to feed a density-increment argument.

Increment chains and $(d,\eta)$-small sequences (Sections 6.1, 6.6)

Logic. Definition 6.6 fixes a $(d,\eta)$-small sequence: same frequency set (rank $d$), nested Bohr sets with radii shrinking so $\nu(B_{i+1}) / \nu(B_i) \le \eta$.

What the tool plots. Radii $\varepsilon_i = \varepsilon_0 \eta^i$ at fixed $\Theta$, with cardinalities $\lvert \Lambda_{\Theta,\varepsilon_i}\rvert$. Dimension stays fixed while the radius collapses — the same “zoom” iteration that supports the density-increment loop (as in the proof structure toward results such as Theorem 7.9 in the paper).

$\ell_{1}$-spread heatmap (Sections 6.1, 6.3; Definition 6.13)

Breakthrough (informal). The diagonal-type object $D$ is not treated with the same algebraic spreadness used for the other sides; the authors use $\ell_{1}$-spreadness (Definition 6.13) as a workable substitute.

Definition 6.13 (as in the paper: $B_1,B_2$ regular Bohr sets, $\nu(B_2) \le \nu(B_1)$). A function $f : B_1 \to [0,1]$ is $(B_1,B_2,\varepsilon)$ $\ell_{1}$-spread when

$$ \mathbb{E}_{x \sim B_1},\Bigl\lvert \mathbb{E}_{y \sim B_2}[f(x+y)] - \mathbb{E}[f] \Bigr\rvert ;\le; \varepsilon \cdot \mathbb{E}[f], $$

with $\mathbb{E}[f] = \mathbb{E}_{x \sim B_1}[f(x)]$.

What the tool computes. For $f = \mathbf{1}_{A}$ (extended by zero off $A$), it approximates the left-hand side above and compares it to $\varepsilon_{\mathrm{tolerance}} \cdot \mathbb{E}[f]$. The bar plot over $x \in \mathbb{Z}_{N}$ shows pointwise magnitudes $\bigl\lvert \mathbb{E}_{y \sim B_2}[f(x+y)] - \mathbb{E}[f]\bigr\rvert$ — empirical feedback for the log-potential style analysis that rules out persistent “dips below the mean” on bad pieces of $D$.

Stability and scaling (Sec. 6.3 · Sec. 7.1)

Nested density ratio. The Scaling · pipe tab compares the empirical nested cardinality ratio $\lvert B_1\rvert/\lvert B_2\rvert$ (from $\varepsilon_{\mathrm{outer}}$ scans at fixed $\varepsilon_{\mathrm{inner}}$ in the expander) to the Lemma 6.2–style guide $C\cdot(\varepsilon_{\mathrm{outer}}/\varepsilon_{\mathrm{inner}})^d$. Constants $C$ are illustrative; the aim is to verify dimension scaling in $\varepsilon$, not sharp constants.

Anisotropic Bohr. An optional $\varepsilon_{\mathrm{XY}}$ / $\varepsilon_{\mathrm{D}}$ split assigns different per-character radii on $\mathbb{T}^d$ (proxy for treating the diagonal direction differently from the $X,Y$ directions in Sec. 7.1).

grid_norm_pipe_v1. The same tab exports the current Bohr indicator mask as JSON (schema: grid_norm_pipe_v1) for downstream Gowers grid norm auditors (structured majorants, Sec. 2.3 style).

Lab components vs paper concepts

Lab component	Paper concept	Purpose
Sub-Bohr sifting	Relative sifting (see Sec. 2.2, Sec. 3.5)	Find structure inside a sparse majorant $B_1$ by measuring density relative to $B_1$ and refinement inside $B_2$.
Increment chain	$(d,\eta)$-small / exact sequences (Def. 6.6)	Model the iterative Bohr “zoom” where rank is fixed and radius shrinks.
$\ell_{1}$-spread analysis	Definition 6.13	Test stability of the diagonal-type direction when full algebraic spreadness is unavailable.
Fourier spectrum (dual group)	Sec. 6.2, Appendix A (almost periodicity)	$\bigl\lvert \widehat{\mathbb{1}_{\Lambda}}(r)\bigr\rvert$ over $r \in \mathbb{Z}_{N}$; compare energy at low modes and at the fixed $\Theta$ to see sparse frequency support.
Balanced strip	Balanced $f - \mathbb{E}[f]$ (Sec. 2, Sec. 4)	Visualize $\mathbb{1}_{\Lambda} - \alpha$ (global or on $B_1$) to see mean-zero “clumpiness” used in Gowers-type arguments.
Pair health map (proxy)	Sec. 7.1 (well-conditioned pairs), Lemma 7.5 (not poor)	Coarse 0–3 score from slice-stability of $X,Y$ and a $D$-tube check; not a full $(B_i,B_8,B_9,K,K)$ grid norm.
Scaling · pipe	Lemma 6.2 (mass in $\varepsilon$), Sec. 7.1 / Sec. 2.3	Nested $\lvert B_1\rvert/\lvert B_2\rvert$ vs $C\cdot(\varepsilon_{\mathrm{outer}}/\varepsilon_{\mathrm{inner}})^d$; optional $\varepsilon_{\mathrm{XY}}/\varepsilon_{\mathrm{D}}$ (anisotropic Bohr); grid_norm_pipe_v1 JSON export.

Presentation note ($\ell_{1}$-spread and the heatmap)

Why check $\ell_{1}$-spreadness on a heatmap? In Section 6.3 the authors use a log-potential argument to show that if a set is not $\ell_{1}$-spread, it can be partitioned into pieces that are (cf. Lemma 6.21 and the surrounding recursion). The heatmap marks residues $x$ where the pointwise term $\bigl\lvert \mathbb{E}_{y \sim B_2}[f(x+y)] - \mathbb{E}[f]\bigr\rvert$ is large — i.e. where the obstructions to spreadness live and where “bad pieces” would be carved out in that partitioning step.

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Figures (generated)

These PNGs live under docs/images/ and are referenced with relative paths so they render on GitHub. Most plots below come from scripts/generate_readme_figures.py; streamlit_app.png (under Streamlit UI) is updated manually from a running app (streamlit run app.py). Regenerate script-driven figures after changing script defaults:

python scripts/generate_readme_figures.py

Core Bohr set views

Cardinality vs radius — $\lvert \Lambda_{\Theta,\varepsilon} \rvert$ for a fixed $\Theta \subset \mathbb{Z}_{180}$:

Membership in $\mathbb{Z}_{N}$ (same parameters):

Regularity-style ratios — sandwich $\Lambda_{\varepsilon(1\pm\sigma)}$ vs $\Lambda_{\varepsilon}$ (qualitative analogue of Definition 6.3):

Fourier dual group (almost periodicity viewpoint)

$\bigl\lvert \widehat{\mathbb{1}_{\Lambda}}(r)\bigr\rvert$ on $\mathbb{Z}_{N}$ (vertical lines at frequencies in $\Theta$):

Conceptual diagrams (from earlier sections)

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Code map

File	Role
bohr_set.py	Torus norm, $\Lambda_{\Theta,\varepsilon}$, regularity sandwich, random walk / TV, sub-Bohr sifting, Bohr chains, $\ell_{1}$-spread, dual Fourier magnitudes, balanced function, pair-health proxy matrix
app.py	Streamlit UI: nested $B_2 \subset B_1$, sparse $A$, Fourier spectrum, balanced strip, coloring strip (Sec. 8 pedagogy), pair-health heatmap, scaling · pipe, earlier tabs
scripts/generate_readme_figures.py	Writes docs/images/*.png (Bohr curves, indicator, regularity, Fourier, quantitative arc, non-abelian pipeline)

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Setup

cd "c:\Users\adith\Bohr Set Visualizer"
pip install -r requirements.txt
streamlit run app.py

Python 3.10+ recommended.

Tests

cd "c:\Users\adith\Bohr Set Visualizer"
pip install -r requirements.txt
$env:PYTEST_DISABLE_PLUGIN_AUTOLOAD = "1"   # avoids broken third-party pytest plugins on some machines
python -m pytest tests/ -v

Or run powershell -File scripts/run_tests.ps1.

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License

No license file is included; add one if you redistribute.