TY - JOUR

T1 - On the additive completion of polynomial sets of integers

AU - Donagi, Ran

AU - Herzog, Marcel

PY - 1971/5

Y1 - 1971/5

N2 - Let f(x) = Σt=0n ctxt be a polynomial of degree n with nonnegative integral coefficients, and let U: a1 < a2 < ... < ak be the set of integers of the form ai = f(i) that belong to the interval [1, N]. Let B: b1 < b2 < ... < bl be an additive completion of U for the interval [1, N]. Then given ε{lunate} > 0, we have kl > (1 + (n - 1) 2n2 - ε{lunate}) N for sufficiently large N. A similar result is also proved under more general conditions, which suffice for the verification of Hanani's conjecture.

AB - Let f(x) = Σt=0n ctxt be a polynomial of degree n with nonnegative integral coefficients, and let U: a1 < a2 < ... < ak be the set of integers of the form ai = f(i) that belong to the interval [1, N]. Let B: b1 < b2 < ... < bl be an additive completion of U for the interval [1, N]. Then given ε{lunate} > 0, we have kl > (1 + (n - 1) 2n2 - ε{lunate}) N for sufficiently large N. A similar result is also proved under more general conditions, which suffice for the verification of Hanani's conjecture.

UR - http://www.scopus.com/inward/record.url?scp=0345949524&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(71)90031-X

DO - 10.1016/0022-314X(71)90031-X

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AN - SCOPUS:0345949524

SN - 0022-314X

VL - 3

SP - 150

EP - 154

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -