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Statement

Setting (§2, p. 8). For non-negative $c_1,\ldots,c_p$ and $1\le p\le n$, the paper's (6) defines

$$N(c_1,\ldots,c_p)=\max_{0<\theta\le2\pi}\sum_{k=1}^{p}c_k \log\bigl|1-e^{ki\theta}\bigr|.$$

When the $c_k$ are non-negative integers with sum $n$, this is $\log M(a_1,\ldots,a_n)$ for the exponents in which each $k$ occurs $c_k$ times.

Lemma 2 (p. 10). Let $c_0,\ldots,c_p$ be non-negative and not all zero, and suppose that

$$\sum_{k=0}^{p}c_k\cos k\phi\ge0\qquad(15)$$

for every real $\phi$. Then for every positive integer $M$,

$$N(c_1,\ldots,c_p)\le c_0\log M+2M^{-1}\sum_{k=1}^{p}c_k\log2.\qquad(16)$$

Source. Lemma 2, stated on p. 10 and proved on p. 11, of F. V. Atkinson, On a problem of Erdős and Szekeres, Canad. Math. Bull. 4 (1961), 7–12, DOI 10.4153/CMB-1961-002-5, as identified on the source card.

Read depth. Claims checked: the setting, hypotheses and conclusion were read clause by clause on pp. 8, 10 and 11, and the short deduction on p. 11 was followed. Nothing here is independently reviewed.

Proof pointer

p. 11. Insert Lemma 1 with the same $M$ into each term of (6). With $j(\phi)=\sum_{k=1}^pc_k\cos k\phi$ (p. 8, (9)), hypothesis (15) gives $-j(m\theta)\le c_0$ for every $m$, and the weights satisfy $\sum_{m=1}^{M-1}(1-m/M)^2m^{-1}\le\log M$; the error terms add up to $M^{-2}(2M-1)\log2\sum_{k=1}^pc_k$, which is at most the second term of (16).

Dependencies

Lemma 1.

Bears on

• Problem 256: the lemma bounds $\log M(a_1,\ldots,a_n)$ when exponent $k$ has multiplicity $c_k$, in terms of any constant term $c_0$ that makes the cosine polynomial with these coefficients non-negative; with the Fejér coefficients it gives inequality (5). On its own it states no bound for $f(n)$.