arXiv:0906.4209v2

On Larcher's theorem concerning good lattice points and multiplicative subgroups modulo p

Nikolay G. Moshchevitin, Dmitrii M. Ushanov

math.NT11K3111K38

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Paper metadata

Original submission: June 23, 2009
Latest revision: August 8, 2009 · version 2
Claimed MSC: 11K31, 11K38
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: The current version (v2) contains a result stronger than that from the version 1

Abstract

We prove the existence of two-dimensional good lattice points in thick multiplicative subgroups modulo $p$ .

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 20, 2026

01Statements2 reported findingsCorrect

The partial-quotient and discrepancy existence results for thick multiplicative cosets are correct.

Theorem 1Correct

Uniform partial-quotient bound on more than half of the coset

Page 1 · Theorem 1 · arXiv:0906.4209v2

For $|R|\geq10^5p^{7/8}(\log p)^{3/2}$ , the character-sum estimate makes the average number of forbidden short congruences less than $1/2$ . Hence more than half of $a\in R$ have none, and the continued-fraction criterion gives $b_j(a)<16\log p$ for every partial quotient.

Theorem 2 and corollaryCorrect

Small total partial quotient and low discrepancy

Pages 1–2 · Theorem 2 and Corollary · arXiv:0906.4209v2

The layer-cake count for the functions $f_a$ and the subgroup character estimate yield an $a\in R$ with $\sum b_i(a)\leq500\log p\log\log p$ . The standard discrepancy–continued-fraction inequality then gives the stated $D_p(a)\ll\log p\log\log p$ .

02Proofs4 reported findingsCorrect

The Burgess estimate, orthogonality calculation, and continued-fraction counting argument are correct; one subgroup symbol is misprinted.

Lemma 1Correct and complete

Character sum over the hyperbolic cover

Pages 2–3 · Lemma 1 · arXiv:0906.4209v2

Each rectangle factors into two interval character sums. Burgess with $r=2$ bounds their product by $900k p^{3/8}\log p$ , and summing over at most $2\log_2p$ rectangles gives the stated $10000p^{7/8}(\log p)^2/\sqrt c$ estimate.

Proof of Theorem 1Correct and complete

Character orthogonality and the continued-fraction criterion

Pages 3–4 · Equations (3)–(6) and proof of Theorem 1 · arXiv:0906.4209v2

Absence of a counted congruence gives $\lVert ax/p\rVert>1/(xt)$ . Lemma B then forces every partial quotient below $t$ . Averaging the nonnegative count over the coset leaves only characters trivial on $U$ ; the principal term is at most $|R|/4$ and Lemma 1 makes the nonprincipal contribution smaller than $|R|/4$ under the stated size hypothesis.

Equation (7)Typo

The coset notation should refer to $U$ and $R=vU$

Page 5 · Equation (7) and the following paragraph · arXiv:0906.4209v2

In the count of $B(c)$ , replace the undefined $K$ in $b\in xK$ by the coset $R$ , and replace “characters trivial on $R$ ” by “characters trivial on $U$ .” Since $a\in R=vU$ , multiplication by $x$ puts the residue in $xR$ , while the orthogonality relation is over the subgroup $U$ . The following factors $\chi(v)$ and $|R|=|U|$ confirm the unique intended notation.

Proof of Theorem 2Correct and complete

Partial summation and averaging

Pages 4–6 · proof of Theorem 2 · arXiv:0906.4209v2

For $a$ from the good half-coset, $f_a(x)<16\log p$ . Summing its level sets reduces the mean of $S_a$ to $\sum_{c\leq t}|B(c)|$ . The principal harmonic sum and the $c^{-1/2}$ nonprincipal sum give the displayed bound, and the hypothesis $|R|\geq10^8p^{7/8}(\log p)^{5/2}$ leaves an element with the required total partial quotient.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:0906.4209v2
Authors listed: Nikolay G. Moshchevitin, Dmitrii M. Ushanov
Audit date: August 20, 2026

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On Larcher's theorem concerning good lattice points and multiplicative subgroups modulo p

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Uniform partial-quotient bound on more than half of the coset

Small total partial quotient and low discrepancy

Character sum over the hyperbolic cover

Character orthogonality and the continued-fraction criterion

The coset notation should refer to UUU and R=vUR=vUR=vU

Partial summation and averaging

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Authenticate with ORCID, then require an exact arXiv match

The coset notation should refer to $U$ and $R=vU$