On some density theorems in number theory and group theory
Bibliographic record
Abstract
Gowers [31] in his paper on quasirandom groups studies a question of Babai and Sos asking whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|. Answering the question negatively, he proves that for sufficiently large prime p, the group PSL2( Fp ) has no product-free subset of size ≥ cn 8/9, where n is the order of PSL2( Fp ). We will consider the problem for compact groups and in particular for the profinite groups SLk( Zp ) and Sp2k( Zp ). In Part I of this thesis, we obtain lower and upper exponential bounds for the supremal measure of the product-free sets. The proof involves establishing a lower bound for the dimension of non-trivial representations of the finite groups SLk( Z /(pn Z )) and Sp2k( Z /(pn Z )). Indeed, our theorem extends and simplifies previous work of Landazuri and Seitz [49], where they consider the minimal degree of representations for Chevalley groups over a finite field. In Part II of this thesis, we move to algebraic number theory. A monogenic polynomial f is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime q, using the Chebotarev density theorem, we will show the density of primes p, such that tq – p is monogenic, is greater than or equal to (q – 1)/q. We will also prove that, when q = 3, the density of primes p, which Qpp3p is non-monogenic, is at least 1/9. Keywords. Profinite group, Complex representation, Hilbert-Schmidt operator, Singular value decomposition, Chebotarev density theorem, Monogenic field, Thue equation.
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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 | 0.005 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.020 | 0.002 |
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; both teacher heads agree on what is shown here.
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".