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More generally, we show that for any $0 < \\alpha \\le k$, if $B \\subset {\\mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${\\mathbb R}^n$ such that every $P \\in E$ intersects $B$ in a set of Hausdorff dimension at least $\\alpha$, then $\\dim B \\ge 2 \\alpha - k + \\min(\\dim E, 1)$, where $\\dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well know"},"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":"1701.02299","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2017-01-09T18:48:48Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"70c7f11c5ebc9e806bccd6fc63624305d71d861dba7abb61ce30f289038d645a","abstract_canon_sha256":"6fec4f6b3455eae06ea9e334ec49d85a863ccd3bd0a6319ca93f3d9b96ac41d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:51.442850Z","signature_b64":"7C927GwUsS9RIpEXluSHyIUzPl+ZmH+1/Ag9nsZ/k4GylaNvAjnSBnSaMmQneKfyX6vD6JPudDXTDua0gFgUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01597fbc48bad9d073506f83525e702a154760b4926072f310a903eb7db8228a","last_reissued_at":"2026-05-18T00:21:51.442204Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:51.442204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"A. M\\'ath\\'e, K. H\\'era, T. Keleti","submitted_at":"2017-01-09T18:48:48Z","abstract_excerpt":"We prove that for any $1 \\le k<n$ and $s\\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < \\alpha \\le k$, if $B \\subset {\\mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${\\mathbb R}^n$ such that every $P \\in E$ intersects $B$ in a set of Hausdorff dimension at least $\\alpha$, then $\\dim B \\ge 2 \\alpha - k + \\min(\\dim E, 1)$, where $\\dim$ denotes the Hausdorff dimension. 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