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Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \\ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and prove a \"dual\" sum-products estimate."},"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":"1508.03814","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-16T11:50:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4ea4d726b1af82be31b81867e58d8aebd7360f6b5a81dd2781ff1505ccae3e45","abstract_canon_sha256":"2f651db7a458219eeeb21cb29947f0f1d6f66d6cc157be61dedcde70e7c38dd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:15.177004Z","signature_b64":"R/4cde54I1vWTXtANXDUfvmPIqcne/JRo/wprtyW2n5CixK183t2v7e60EkeyKKdkL6fxWjreTMbRTgZ5EzjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6712ed22f0655c63aa336c649a40c334c25a65c1b5b5575a0cb7327c7063f67","last_reissued_at":"2026-05-18T01:35:15.176635Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:15.176635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Differences of subgroups in subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ilya D. 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