Equal sums in random sets and the concentration of divisors
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.083 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.007 | 0.004 |
| Science and technology studies | 0.003 | 0.010 |
| Scholarly communication | 0.008 | 0.011 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".