TY - JOUR

T1 - Exact limit theorems for restricted integer partitions

AU - Antonir, Asaf Cohen

AU - Shapira, Asaf

N1 - Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/10/8

Y1 - 2022/10/8

N2 - For a set of positive integers A, let pA(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy–Ramanujan formula for p(n) by showing that A has density α if and only if logpA(n)∼logp(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if logpA(n)/logp(αn) has lower limit 1. We answer this question negatively by constructing, for every α>0, a set of integers A of lower density α, satisfying [Formula presented] We further show that the above bound is best possible (up to the oα(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

AB - For a set of positive integers A, let pA(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy–Ramanujan formula for p(n) by showing that A has density α if and only if logpA(n)∼logp(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if logpA(n)/logp(αn) has lower limit 1. We answer this question negatively by constructing, for every α>0, a set of integers A of lower density α, satisfying [Formula presented] We further show that the above bound is best possible (up to the oα(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

KW - Elementary proofs

KW - Number theory

KW - Partition function

UR - http://www.scopus.com/inward/record.url?scp=85134312128&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2022.108554

DO - 10.1016/j.aim.2022.108554

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AN - SCOPUS:85134312128

SN - 0001-8708

VL - 407

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108554

ER -