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Then \\[\\Bigl \\lvert\\Bigl\\{(a,b)\\in\\bigl\\{0,\\ldots,2^k-2\\bigr\\}^2:a+b\\equiv t\\bmod 2^k-1, w(a)+w(b)<k\\Bigr\\}\\Bigr \\rvert\\leq 2^{k-1}.\\]\n  We prove that the Tu--Deng Conjecture holds almost surely in the following sense: the proportion of $t\\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\\rightarrow\\infty$.\n  Moreover, we prove that the Tu--Deng Conjectu"},"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":"1707.07945","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-07-25T12:12:35Z","cross_cats_sorted":["cs.CR","math.NT"],"title_canon_sha256":"4100f6f20d8aa208ac614a5189a4855e9f085418e8eb73db5a2b14a7f63ff3fd","abstract_canon_sha256":"c50c82b9e5b7e9048dc2439209f95a63fa4b303ece6bfa097d4ce26ba72321c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:00.496420Z","signature_b64":"VwVM5jrqV049fPmnOKwti/awVplLoJBK0MkQ2NJcJ+T3f0z01kGkneOZ9TfBpA9R0ujzA+iVp0dllbLLnccdDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"717aa09ed73e281d2c4bde33ff8a8e900529abc3a39f8d4b4a27ead08ea293ef","last_reissued_at":"2026-05-18T00:11:00.495686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:00.495686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Tu--Deng Conjecture holds almost surely","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","math.NT"],"primary_cat":"math.CO","authors_text":"Lukas Spiegelhofer, Michael Wallner","submitted_at":"2017-07-25T12:12:35Z","abstract_excerpt":"The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\\leq t<2^k-1$. 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