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We show that if $(X,\\mathcal{B},\\mu, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$ $1\\leq i\\leq 6$ are bounded functions then the averages\n  $$\\frac{1}{N^3}\\sum_{n, m, p=1}^N f_1(T_1^nx) f_2(T_2^mx) f_3(T_3^px) f_4(T_4^{n+m}x) f_5(T_5^{n+p}x) f_6(T_6^{m+p}x)$$\n converge almost everywhere."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0511062","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2005-11-02T19:45:06Z","cross_cats_sorted":[],"title_canon_sha256":"3a38c48bd3de6a3a482e7c2b5919451d542d1e2147552b1645396c669b6c33b7","abstract_canon_sha256":"2ade10ecdcbcf792d2c7f4edf902b3a064c62ddaf762925e4afd5a9044a710aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:22.911947Z","signature_b64":"TytLWWelnHv6P26HVm+yRZ/15OBZdvIG5wbvKCSo2Gh+xu5MbZzNr9F1s/snAcfY2unNkL0WHvWrGHYMrxODDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"421e2a116e18e8beacb5adf0b4359eba9b9d69682a59bbc42c7d951f634e5180","last_reissued_at":"2026-05-18T01:05:22.911502Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:22.911502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Averages along cubes for not necessarily commuting measure preserving transformations","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Idris Assani","submitted_at":"2005-11-02T19:45:06Z","abstract_excerpt":"We study the pointwise convergence of some weighted averages linked to averages along cubes. 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