[Merged by Bors] - feat(Analysis/BoxIntegral/UnitPartition): Prove results linking integral point counting and integrals #12405
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UnitPartition
PR summary f021d1f602Import changes exceeding 2%
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| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.Analysis.BoxIntegral.UnitPartition | 1895 | 1980 | +85 (+4.49%) |
Import changes for all files
| Files | Import difference |
|---|---|
Mathlib.Analysis.BoxIntegral.UnitPartition |
85 |
Declarations diff
+ ContinuousOn.continuousAt_mulIndicator
+ _root_.tendsto_card_div_pow_atTop_volume
+ _root_.tendsto_card_div_pow_atTop_volume'
+ _root_.tendsto_tsum_div_pow_atTop_integral
+ eqOn_mulIndicator'
+ eq_of_mem_smul_span_of_index_eq_index
+ integralSum_eq_tsum_div
+ mem_smul_span_iff
+ repr_isUnitSMul
+ setFinite_inter
+ smul
+ tag_index_eq_self_of_mem_smul_span
+ tag_mem_smul_span
You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>
## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>The doc-module for script/declarations_diff.sh contains some details about this script.
No changes to technical debt.
You can run this locally as
./scripts/technical-debt-metrics.sh pr_summary
- The
relativevalue is the weighted sum of the differences with weight given by the inverse of the current value of the statistic. - The
absolutevalue is therelativevalue divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).
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@MichaelStollBayreuth, I think I have addressed all of your comments. Thanks a lot for your extensive review! |
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🚀 Pull request has been placed on the maintainer queue by MichaelStollBayreuth. |
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…ral point counting and integrals (#12405) We prove the following result: > Let `s` be a bounded, measurable set of `ι → ℝ` whose frontier has zero volume and let `F` be a > continuous function. Then the limit as `n → ∞` of `∑ F x / n ^ card ι`, where the sum is over the > points in `s ∩ n⁻¹ • (ι → ℤ)`, tends to the integral of `F` over `s`. using Riemann integration. As a special case, we deduce that > The limit as `n → ∞` of `card (s ∩ n⁻¹ • (ι → ℤ)) / n ^ card ι` tends to the volume of `s`. Both of these statements are for a variable `n : ℕ`. However, with the additional hypothesis: `x • s ⊆ y • s` whenever `0 < x ≤ y`, we generalize the previous statement to a real variable. This PR is part of the proof of the Analytic Class Number Formula. Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com>
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…ral point counting and integrals (#12405) We prove the following result: > Let `s` be a bounded, measurable set of `ι → ℝ` whose frontier has zero volume and let `F` be a > continuous function. Then the limit as `n → ∞` of `∑ F x / n ^ card ι`, where the sum is over the > points in `s ∩ n⁻¹ • (ι → ℤ)`, tends to the integral of `F` over `s`. using Riemann integration. As a special case, we deduce that > The limit as `n → ∞` of `card (s ∩ n⁻¹ • (ι → ℤ)) / n ^ card ι` tends to the volume of `s`. Both of these statements are for a variable `n : ℕ`. However, with the additional hypothesis: `x • s ⊆ y • s` whenever `0 < x ≤ y`, we generalize the previous statement to a real variable. This PR is part of the proof of the Analytic Class Number Formula. Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com>
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…ral point counting and integrals (#12405) We prove the following result: > Let `s` be a bounded, measurable set of `ι → ℝ` whose frontier has zero volume and let `F` be a > continuous function. Then the limit as `n → ∞` of `∑ F x / n ^ card ι`, where the sum is over the > points in `s ∩ n⁻¹ • (ι → ℤ)`, tends to the integral of `F` over `s`. using Riemann integration. As a special case, we deduce that > The limit as `n → ∞` of `card (s ∩ n⁻¹ • (ι → ℤ)) / n ^ card ι` tends to the volume of `s`. Both of these statements are for a variable `n : ℕ`. However, with the additional hypothesis: `x • s ⊆ y • s` whenever `0 < x ≤ y`, we generalize the previous statement to a real variable. This PR is part of the proof of the Analytic Class Number Formula. Co-authored-by: Xavier Roblot <46200072+xroblot@users.noreply.github.com>
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We prove the following result:
using Riemann integration. As a special case, we deduce that
Both of these statements are for a variable
n : ℕ. However, with the additional hypothesis:x • s ⊆ y • swhenever0 < x ≤ y, we generalize the previous statement to a real variable.This PR is part of the proof of the Analytic Class Number Formula.