Published paper
Abstract
Schmidt gives an independent Siegel-style proof of the mean-value formula for linearly independent lattice tuples and of Rogers’ full mean-value formula for arbitrary nonnegative Borel functions when the number of vectors is smaller than the ambient dimension. This is the proof repair needed for the focal primitive second-moment and cusp-volume calculation.
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:
- Satz 2–3 and Lemma 4 · Monatshefte für Mathematik 61 (1957), printed pp. 273–276; equations (8)–(11), Satz 2–3, and Lemma 4Satz 2 proves the independent-tuple mean formula for arbitrary nonnegative Borel functions, and Satz 3 plus Lemma 4 proves the complete Rogers mean-value formula for k<n. Substituting this valid input into the elementary primitive-vector decomposition proves the focal first and second moments in D=m+n≥3.
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Exact reviewed source
Monatshefte für Mathematik version of record · volume 61 (1957), pages 269–276
Wolfgang Schmidt. Mittelwerte über Gitter. Monatshefte für Mathematik 61 (1957), 269–276.
Open audited source ↗01Statements4 reported findingsCorrect
The parabolic mean-value identity, the independent-tuple formula, and Schmidt's Rogers expansion are correct at their stated scope. In particular, Satz 2, Satz 3, and Lemma 4 provide a valid replacement for the analytic part of Rogers's mean-value proof for every nonnegative Borel function when , with extended value allowed.
Satz 1Correct
The parabolic one-vector mean-value formula and deformation consequence are correct
Printed pages 270–273 · equations (4)–(7) and Satz 1
Averaging first over the compact unipotent quotient and then over the lower-right quotient reduces the sum over integer vectors with nonzero last- projection to the ordinary Siegel integral. The product measure is normalized in equation (6), and nonnegativity permits every exchange of sum and integral. The stated deformation consequence uses the case and a determinant-one diagonal scaling; the vectors in the exceptional coordinate plane leave the fixed compact support, after which the average supplies the required lattice.
Schmidt 1957 version of record, printed pages 270–273 ↗Satz 2Correct
The independent-tuple identity holds for every
Printed pages 273–274 · Satz 2 and equations (8)–(10)
For , the set of integer -tuples whose last- projections are independent, an integral unimodular change of coordinates places the first projections in a coordinate subspace. The admissible last vector is then exactly a vector outside the complementary coordinate plane. Satz 1 evaluates that inner sum, and induction on yields Taking gives the full independent-tuple formula for .
Schmidt 1957 version of record, printed pages 273–274 ↗Satz 3Correct
The Rogers expansion is valid for arbitrary nonnegative Borel input when
Printed pages 274–276 · Satz 3, Lemma 3, and Lemma 4
Rogers's combinatorial Lemma 3 partitions every integer -tuple uniquely by its rank, a division of the column indices, a denominator , and an integral matrix . Schmidt averages each rank- stratum using Satz 2 and Lemma 4. Because all terms are nonnegative, Tonelli's theorem justifies the countable rearrangement without a separate convergence hypothesis, so equality remains meaningful when both sides are infinite. This proves the advertised formula independently of Rogers's defective fundamental-domain transfer step.
Schmidt 1957 version of record, printed pages 274–276 ↗Scope boundaryCorrectly scoped
The restrictions to nonnegative functions and are explicit and essential to this theorem
Printed pages 269 and 275 · introduction and Satz 3
The paper does not assert the formula for signed input without absolute-integrability hypotheses, nor for . Its proof uses nonnegativity for Tonelli and Satz 2 with rank . For bounded compactly supported input, the 1957 paper only points to the forthcoming convergence proof; Schmidt's 1958 paper supplies that finiteness result for the full valid range . None of these boundaries weakens the extended-value Borel identity actually stated here.
Schmidt, On the convergence of mean values over lattices (1958), Theorem 2 ↗02Proofs4 reported findingsCorrect
The proofs close at their stated scope. The crucial replacement is genuinely independent at the analytic level: Schmidt proves the independent-tuple mean value through parabolic quotients and uses Rogers only for a finite-dimensional combinatorial decomposition. No step imports Rogers's invalid fundamental-domain invariance argument.
Borel extensionCorrect via the cited approximation argument
The base Siegel identity is available for arbitrary nonnegative Borel functions
Printed page 270 · discussion following equation (4)
Schmidt starts from Siegel's formula for bounded compactly supported Riemann-integrable functions, records the bounded-Borel approximation, and then passes to arbitrary nonnegative Borel functions by truncation, allowing divergence. The local exposition is abbreviated but explicitly points to the detailed approximation on Rogers's printed pages 270–273. That cited measure-theoretic step requires only the already valid Siegel identity for the approximants; it does not use the faulty transfer argument in Rogers's proof of his higher-rank formula.
Rogers 1955 version of record, printed pages 270–275 · Borel extension of Theorem 3 ↗Proof of Satz 2Correct and complete
The induction and quotient disintegration have the required invariance
Printed pages 273–274 · equations (9)–(10)
The integer normal-form matrix belongs to the discrete subgroup, so replacing by preserves the quotient integral. The nested parabolic subgroup used for the last vector has normalized quotient mass one, equation (6) supplies the disintegration, and Satz 1 applies to the remaining nonzero projection. Repetition removes one vector at a time and terminates at the stated Euclidean integral. The hypothesis guarantees that the complementary projection dimension is positive at every stage.
Schmidt 1957 version of record, printed pages 273–274 ↗Lemma 4Correct and complete
The lattice-index and Jacobian computation give the printed coefficient
Printed page 276 · Lemma 4
The congruence solutions form a sublattice with index, hence determinant, . Writing converts the constrained independent tuples bijectively into unconstrained independent tuples. The same -variable linear change acts in each of the coordinate rows, so its Jacobian is , exactly the factor in Lemma 4 and Satz 3.
Schmidt 1957 version of record, printed page 276 ↗Independence of the repairCorrect dependency boundary
The only imported Rogers result is the correct combinatorial rank decomposition
Printed pages 275–276 · Lemma 3 and its use in Satz 3
Schmidt cites Rogers's equations (36)–(37) for the one-to-one decomposition of a lattice -tuple into rank data. Rogers proves that lemma algebraically by choosing the first independent columns, clearing the uniquely determined rational coefficients, and checking the inverse correspondence. The lemma is separate from the later integration step. Schmidt's Satz 2 and Lemma 4 replace that integration step, so the proof does not inherit the invalid right-translate argument identified in Rogers's Theorem 1 proof.
Rogers 1955 version of record, printed pages 275–278 · equations (36)–(42) ↗03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.