arXiv:2503.15175v4
Decomposition results for multiplicative actions and applications
math.DSmath.COmath.NT37A4405D1011B3011N3737A15
Abstract
Motivated by partition regularity problems of homogeneous quadratic equations, we prove multiple recurrence and convergence results for multiplicative measure preserving actions with iterates given by rational sequences involving polynomials that factor into products of linear forms in two variables. We focus mainly on actions that are finitely generated, and the key tool in our analysis is a decomposition result for any bounded measurable function into a sum of two components, one that mimics concentration properties of pretentious multiplicative functions and another that mimics vanishing properties of aperiodic multiplicative functions. Crucial to part of our arguments are some new seminorms that are defined by a mixture of addition and multiplication of the iterates of the action, and we prove an inverse theorem that explicitly characterizes the factor that these seminorms naturally determine.
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01Statements3 reported findingsNo statement issues found
The decomposition, mixed-seminorm inverse theorem, and the convergence and recurrence results for finitely generated multiplicative actions have the scope stated. The extension to one arbitrary action in the degree-zero recurrence case retains every necessary non-power and normalization hypothesis.
Theorems 2.3 and 2.4No issue found
The pretentious–aperiodic decompositions have the claimed consequences
PDF pp. 6–7 and 23–30 · arXiv:2503.15175v4
The spectral support projection defines the pretentious factor, and the concentration estimates apply on that support. Orthogonality to the factor puts the complementary spectral measure on aperiodic multiplicative functions. The cited uniformity and Daboussi–Kátai inputs then give vanishing for every permitted product of independent linear factors, while the stronger mixed-seminorm conclusion is asserted only for finitely generated actions.
Theorems 2.1 and 6.3No issue found
The mixed-seminorm control yields the several-linear-forms theorem
PDF pp. 4–5 and 25–34 · arXiv:2503.15175v4
Repeated van der Corput reduction bounds the multilinear averages by a mixed seminorm. Finite generation supplies a divergent prime class on which the prime transformations agree, and the inverse theorem identifies vanishing of all mixed seminorms with the aperiodic spectral component. Concentration of the pretentious components on residue grids gives a Cauchy approximation for convergence and, together with the conditional-expectation inequality, the stated positive-density recurrence bound.
Theorem 2.2; Propositions 8.5–8.8No issue found
The two-rational-polynomial conclusions follow under the primitive exponent hypotheses
PDF pp. 5–6 and 36–44 · arXiv:2503.15175v4
An invertible linear change sends the two factors of the second iterate to coordinate monomials. The two-dimensional orthogonality criterion then cancels that iterate on a divergent prime class, while the non-power lemma leaves an aperiodic first iterate to which the polynomial vanishing theorem applies. The finitely generated concentration proposition proves convergence and the two recurrence alternatives. In degree zero, restricting to the positive-density sectors where the first ratio is near one gives the required concentration for an arbitrary first action.
02Proofs3 reported findingsNo proof issues found
The spectral, concentration, seminorm, coordinate-change, and sector arguments form a coherent proof chain. Version 4 explicitly handles signed transformed forms, affine congruence restrictions, and the sector normalization used in the general-action recurrence argument.
Propositions 5.2–5.7No issue found in the reviewed proof
The spectral factor and aperiodic vanishing steps are justified
PDF pp. 20–25 · arXiv:2503.15175v4
Bochner–Herglotz represents multiplicative correlations by spectral measures on the character space. Restriction to the Borel pretentious set gives an invariant orthogonal projection and hence a factor. The concentration estimates pass through spectral integration, and the aperiodic characterization combines with the cited uniform polynomial mean-value theorem to prove the required convex-set and additive-character vanishing statements.
Lemma 8.4 and Proposition 8.5No issue found in the reviewed proof
The characteristic-factor reduction preserves the non-perfect-power obstruction
PDF pp. 35–38 · arXiv:2503.15175v4
For each candidate exponent, choosing the smallest nonzero root whose multiplicity is not divisible by that exponent exhibits a factor of the dilation quotient with nondivisible multiplicity. This proves the quotient cannot be a constant times a perfect power. Grid indicators become finite character sums after the linear change, and intersections of their dilates remain convex on the positive support, so the aperiodic vanishing proposition applies exactly as used.
Proposition 8.8; Theorem 2.2(iii)No issue found in the reviewed proof
The degree-zero sector concentration closes the general-action recurrence proof
PDF pp. 42–44 · arXiv:2503.15175v4
The near-level set of a nonconstant homogeneous ratio is, apart from finitely many rays, a finite union of convex sectors and has positive density. Pretentious characters concentrate there because the total homogeneity is zero: the powers of the Følner dilation and the common exponential correction cancel, while the remaining Archimedean phase is uniformly close to one. For each fixed sector width the normalization is bounded below, permitting the aperiodic replacement; the successive limits then recover the conditional-expectation lower bound.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.