A Generalised abc Conjecture and Quantitative Diophantine Approximation
Bibliographic record
Abstract
The abc Conjecture and its number field variant have huge implications across a wide \nrange of mathematics. While the conjecture is still unproven, there are a number of \npartial results, both for the integer and the number field setting. Notably, Stewart \nand Yu have exponential abc bounds for integers, using tools from linear forms in \nlogarithms, while Győry has exponential abc bounds in the number field \ncase, using methods from S-unit equations [20]. In this thesis, we aim to combine \nthese methods to give improved results in the number field case. These results are \nthen applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc \nconjecture. \n \nThe smooth abc conjecture concerns counting the number of solutions to a+b = c \nwith restrictions on the values of a, b and c. this leads us to more general methods \nof counting solutions to Diophantine problems. Many of these results are asymptotic \nin nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these \nlemmas effective rather than asymptotic other than on a set of size δ > 0, where δ is \narbitrary. From there, we apply these tools to give an effective Schmidt’s Theorem, \na quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin- \nSchaeffer Theorem), and to give effective results on inhomogeneous Diophantine \nApproximation on M0-sets, normal numbers and give an effective Strong Law of \nLarge Numbers. We conclude this thesis by giving general versions of Lemmas 1.4 \nand 1.5 of Harman's "Metric Number Theory".
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".