An old conjecture of Erdos–Turán on additive bases

Borwein, Peter; Choi, Stephen; Chu, Frank

doi:10.1090/S0025-5718-05-01777-1

An old conjecture of Erdos–Turán on additive bases
HTML articles powered by AMS MathViewer

by Peter Borwein, Stephen Choi and Frank Chu PDF

Math. Comp. 75 (2006), 475-484

Abstract:

There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate: Conjecture 0.1(Erdős and Turán). Suppose that $0 = \delta _0<\delta _1<\delta _2<\delta _3\cdots$ is an increasing sequence of integers and \[ s(z) : = \sum _{i=0}^\infty z^{\delta _i}. \] Suppose that \[ s^2(z) := \sum _{i=0}^\infty b_i z^i. \] If $b_i>0$ for all $i$, then $\{b_n\}$ is unbounded. Our main purpose is to show that the sequence $\{b_n\}$ cannot be bounded by $7$. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

References

Martin Dowd, Questions related to the Erdős-Turán conjecture, SIAM J. Discrete Math. 1 (1988), no. 1, 142–150. MR 936616, DOI 10.1137/0401016
P. Erdős and R. Frued, On Sidon-sequences and related problems, Mat. Lapok (New Ser.) (1991/2 (in Hungarian)), no. 1, 1–44.
P. Erdös and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. MR 6197, DOI 10.1112/jlms/s1-16.4.212
P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979), no. 3-4, 325–344 (1980). MR 570317
G. Grekos, L. Haddad, C. Helou, and J. Pihko, On the Erdős-Turán conjecture, J. Number Theory 102 (2003), no. 2, 339–352. MR 1997795, DOI 10.1016/S0022-314X(03)00108-2
Melvyn B. Nathanson, Unique representation bases for the integers, Acta Arith. 108 (2003), no. 1, 1–8. MR 1971077, DOI 10.4064/aa108-1-1
Csaba Sándor, Range of bounded additive representation functions, Period. Math. Hungar. 42 (2001), no. 1-2, 169–177. MR 1832703, DOI 10.1023/A:1015261010544