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Let $\\displaystyle s_{A}(x) = \\max_{n \\le x}R_{A}(n)$ and $\\displaystyle d_{A,B}(x) = \\max_{\\hbox{t: $a_{t} \\le x$ or $b_{t} \\le x$}}|a_{t} - b_{t}|$. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. We improve a result of Haddad and Helou about the Erd\\H{o}s - Tur\\'an conjecture."},"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":"1504.07411","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-28T10:17:59Z","cross_cats_sorted":[],"title_canon_sha256":"2a6c825263abd2725091918c7404dd46d9f403add70720e64fdb47d116e2d644","abstract_canon_sha256":"3282218e86c3ef1d0c14398dcab2365889c93969c9e0a2d6587ad16b1481b2f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:47.856437Z","signature_b64":"pJgJDIkzWRaDOOVbQEc/DdFCNI3AGIRX1hX6PUr51kOfAvp2ksQhzw8h7IEm1p7Gn0rHlvCjWo0LxmQxkYkXDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93825972c98233add1b8e1a4da3b5bd283e46e0775c3ecd3d074ed94eb3cefb6","last_reissued_at":"2026-05-18T01:36:47.855766Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:47.855766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the maximum values of the additive representation functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Csaba S\\'andor, S\\'andor Z. Kiss","submitted_at":"2015-04-28T10:17:59Z","abstract_excerpt":"Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\\displaystyle s_{A}(x) = \\max_{n \\le x}R_{A}(n)$ and $\\displaystyle d_{A,B}(x) = \\max_{\\hbox{t: $a_{t} \\le x$ or $b_{t} \\le x$}}|a_{t} - b_{t}|$. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. 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