On Gaps Between Sums of Powers and Other Topics in Number Theory and Combinatorics
Bibliographic record
Abstract
One main goal of this thesis is to show that for every K it is possible to find K consecutive natural numbers that cannot be written as sums of three nonnegative cubes. Since it is believed that approximately 10% of all natural numbers can be written in this way, this result indicates that the sums of three cubes distribute unevenly on the real line. These sums have been studied for almost a century, in relation with Waring's problem, but the existence of ``arbitrarily long gaps'' between them was not known. We will provide two proofs for this theorem. The first is relatively elementary and is based on the observation that the sums of three cubes have a positive bias towards being cubic residues modulo primes of the form p=1+3k. Thus, our first method to find consecutive non-sums of three cubes consists in searching them among the natural numbers that are non-cubic residues modulo ``many'' primes congruent to 1 modulo 3. Our second proof is more technical: it involves the computation of the Sato-Tate distribution of the underlying cubic Fermat variety {x^3+y^3+z^3=0}, via Jacobi sums of cubic characters and equidistribution theorems for Hecke L-functions of the Eisenstein quadratic number field Q(\sqrt{-3}). The advantage of the second approach is that it provides a nearly optimal quantitative estimate for the size of gaps: if N is large, there are >>\sqrt{log N}/(log log N)^4 consecutive non-sums of three cubes that are less than N. According to probabilistic models, an optimal estimate would be of the order of log N / log log N. In this thesis we also study other gap problems, e.g. between sums of four fourth powers, and we give an application to the arithmetic of cubic and biquadratic theta series. We also provide the following additional contributions to Number Theory and Combinatorics: a derivation of cubic identities from a parameterization of the pseudo-automorphisms of binary quadratic forms; a multiplicity estimate for multiprojective Chow forms, with applications to Transcendental Number Theory; a complete solution of a problem on planar graphs with everywhere positive combinatorial curvature.
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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 | 0.003 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".