Equal sums in random sets and the concentration of divisors
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Bibliographic record
Abstract
Abstract We study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>max</mml:mo> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>#</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo>|</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mo>log</mml:mo> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⩾</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>0.35332277</mml:mn> <mml:mo>…</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for almost all n , a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $${\textbf{A}} \subset {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> by selecting i to lie in $${\textbf{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> with probability 1/ i . What is the supremum of all exponents $$\beta _k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> such that, almost surely as $$D \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∩</mml:mo> <mml:mo>[</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> in k different ways? We characterise $$\beta _k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , and obtain lower bounds for $$\beta _k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> which we believe to be asymptotically sharp.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it