Published paper

Mean values over the space of lattices

C. A. Rogers

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Publication metadata

Journal citation: Acta Mathematica 94, 249–287 (1955)
Publication date: 1955
DOI: 10.1007/BF02392493 ↗

Abstract

Theorems 4 and 5 give the primitive first- and second-moment formulas needed for the cusp-volume asymptotic used without a valid citation on focal page 28.

Role in dependence graphs

Proof-critical source

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

This paper is included only for the following marked statement:

Theorem 4 at k=1 and Theorem 5 · Acta Math. printed pp. 251–253 and proof pp. 279–284Primitive first and second moment formulas for lattice-point counts; applied to the sup-norm cube they determine the small-cusp volume.

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Acta Mathematica version of record · volume 94 (1955), pages 249–287

C. A. Rogers. Mean values over the space of lattices. Acta Mathematica 94 (1955), 249–287.

Open audited source ↗

Generated August 23, 2026

01Statements3 reported findingsContains wrong statements

The mean-value formulas are correct in their valid range, and the primitive first/second moments needed by the focal dependence graph are correct in every dimension at least three. Theorem 5 is nevertheless false as stated because it includes dimension two. Theorem 6 remains true in dimension two by Schmidt's later independent argument.

Theorem 5Incorrect in the stated dimension-two case

The primitive two-point formula is false when $n=2$

Printed page 252 · Theorem 5 · equation (12)

The theorem claims equation (12) for every $n\geq2$ . In dimension two, choose a nonnegative Borel function $\rho$ supported in a sufficiently small neighborhood of $(e_1,\tfrac12 e_2)$ , with support disjoint from the loci $X_2=\pm X_1$ and with $\det(X_1,X_2)$ bounded away from every integer. For any unimodular lattice $\Lambda\subset\mathbb R^2$ , the determinant of any two lattice vectors is an integer, so the entire lattice sum on the left of (12) is zero. The double integral on the right is positive and the two diagonal integrals vanish, a contradiction. The valid local repair is to replace $n\geq2$ by $n>2$ , exactly as recorded by Schmidt in 1958.

W. M. Schmidt, On the convergence of mean values over lattices, footnote 2 ↗

Theorem 6Correct · dimension-two proof replaced

The metric finiteness/divergence conclusion is correct in all stated dimensions

Printed pages 253 and 283–287 · Theorem 6

For $n>2$ , the first-moment and primitive second-moment bounds yield the stated almost-everywhere convergence/divergence dichotomy. The printed dimension-two divergence proof is not valid because it invokes the false dimension-two case of Theorem 5, but Schmidt's Theorem 2 supplies a stronger discrepancy estimate in $\mathbb R^2$ and its corollary gives infinitely many primitive points for every nested Borel family of infinite volume. Thus the conclusion is correct although its printed proof is not.

W. M. Schmidt, A metrical theorem in geometry of numbers, Theorem 2 and Corollary ↗

Theorem 5 for $n\geq3$Correct in the focal range

The focal primitive first- and second-moment specialization is correct

Printed pages 251–252 and 279–282 · Theorems 4–5

Let $D\geq3$ and let $N_r$ count primitive vectors of a unimodular lattice in the symmetric cube $(-r,r)^D$ . The primitive Siegel formula and Theorem 5 give $\mathbb E N_r=\frac{2^D r^D}{\zeta(D)},\qquad \mathbb E N_r^2=(\mathbb E N_r)^2+2\mathbb E N_r.$ Since $N_r$ is even, the first and second moments imply $\mu\{\delta(\Lambda)<r\}=\frac{2^{D-1}}{\zeta(D)}r^D+O_D(r^{2D}).$ Subtracting the same estimate at $r/e$ gives the fixed-ratio cusp-annulus bound required by the BFK graph. The 1955 proof of the moment formula still inherits the invalid page-256 transfer step through Theorem 3, but Schmidt's independent 1957 Satz 2–3 and Lemma 4 prove the full arbitrary-Borel formula for $k<D$ . Thus the focal $D=m+n\geq3$ cube specialization is valid despite the adverse whole-paper proof status.

W. Schmidt, Mittelwerte über Gitter, Satz 2–3 and Lemma 4 ↗

02Proofs3 reported findingsContains incorrect or incomplete proofs

Two substantive defects occur. The fundamental-domain argument on printed page 256 uses a false invariance step; later proofs of the Rogers formula replace it. The proof of Theorem 5 applies Theorem 3 outside its range when $n=2$ , producing the false statement above and invalidating the printed dimension-two branch of Theorem 6.

Theorem 1 transfer stepIncorrect as written · verified replacement proofs

The asserted right-translate decomposition of the fundamental domain is invalid

Printed pages 255–258 · especially page 256 and equation (18)

After changing variables, the proof asserts that $F^*O^{-1}$ can be cut into finitely many pieces and returned to $F^*$ by right multiplication with integral unimodular matrices for an arbitrary determinant-one matrix $O$ . That is not a property of a right fundamental domain: right translation conjugates the lattice subgroup. Kim gives an explicit failure using a compactly supported function away from the cusp and $O=\operatorname{diag}(a,a^{-1})$ for large $a$ . This breaks the printed derivation of (18), on which the subsequent transfer to invariant measure rests. Repair classification: Verified repair. Schmidt supplied an alternative proof of Rogers' theorem, and Kim's Hecke-operator argument supplies a later direct replacement for the Rogers integral formula.

Seungki Kim, Mean value formulas on sublattices and flags of the random lattice, arXiv:2005.10874v3, Section 2 ↗

Proof of Theorem 5Incorrect as written · statement repaired only by restricting the dimension

The dimension-two case invokes Theorem 3 outside its hypotheses

Printed pages 279–282 · equations (49)–(53)

The dominated-convergence step (49) and the evaluation leading to (53) invoke Theorem 3 for two linearly independent vectors. Theorem 3 assumes the number of vectors $m$ satisfies $m\leq n-1$ , so it is unavailable when $m=n=2$ . This is not a missing routine endpoint: in dimension two the resulting equation (12) is false by the determinant-support counterexample in the Statements dimension. Repair classification: Verified scope repair $n>2$ ; there is no repair preserving the printed $n=2$ formula. For $D\geq3$ this endpoint problem disappears, although Schmidt's independent proof is still needed to replace the separate transfer defect inherited by Theorem 3.

Rogers version of record and Schmidt's independent 1957 repair ↗

Proof of Theorem 6 in dimension twoIncorrect as written · verified independent repair

The printed second-moment argument inherits the false endpoint

Printed pages 285–287 · divergent-integral branch of Theorem 6

The displayed variance identity is obtained by applying Theorem 5 to the truncated function $ho_R$ . For $n=2$ that input is false, so the contradiction argument does not prove the stated conclusion. Repair classification: Verified repair. Schmidt's separate planar determinant-integral argument proves a quantitative primitive discrepancy bound and hence the dimension-two conclusion of Theorem 6 without using equation (12).

Mean values over the space of lattices

Publication metadata

Abstract

Role in dependence graphs

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

Audit summary

Acta Mathematica version of record · volume 94 (1955), pages 249–287

The primitive two-point formula is false when n=2n=2n=2

The metric finiteness/divergence conclusion is correct in all stated dimensions

The focal primitive first- and second-moment specialization is correct

The asserted right-translate decomposition of the fundamental domain is invalid

The dimension-two case invokes Theorem 3 outside its hypotheses

The printed second-moment argument inherits the false endpoint

The primitive two-point formula is false when $n=2$