Bounds on ternary cyclotomic coefficients

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Bartłomiej Bzdęga, "Bounds on ternary cyclotomic coefficients," Acta Arithmetica, 144(1), 5-16, 2010. https://doi.org/10.4064/aa144-1-2

Summary

For distinct primes $p<q,r$, write $\Phi_{pqr}(x)=\sum_n a_{pqr}(n)x^n$ and let $A_+$, $A_-$, and $A=\max\{A_+,-A_-\}$ be the extremal coefficients defined in (1.1). If $q',r'$ are the inverses of $q,r$ modulo $p$, set

$$\alpha=\min\{q',r',p-q',p-r'\}, \qquad \alpha\beta qr\equiv1\pmod p,\qquad 0<\beta<p.$$

Theorem 1.3 gives the asymmetric estimates

$$A_+\leq\min\{2\alpha+\beta,p-\beta\}, \qquad -A_-\leq\min\{p+2\alpha-\beta,\beta\}.$$

With $\beta^*=\min\{\beta,p-\beta\}$, Theorem 1.4 combines these into $A\leq\min\{2\alpha+\beta^*,p-\beta^*\}$, improving Bachman's bound (1.3), strictly precisely when $\alpha+\beta^*<(p-1)/2$. The estimates depend only on the residue classes of $q$ and $r$ modulo $p$. Section 4 applies them to several regimes: Corollary 4.1 gives explicit congruence classes with $A\leq\min\{2i+j,i+2j\}\leq18$ (and in particular the introduction notes $A\leq3$ when $q,r\equiv\pm1\pmod p$); Corollary 4.2 proves the stated piecewise lower bound for the density $D_p(c)$ and yields the modified Beiter bound $A\leq2p/3$ for at least $25/27+O(1/p)$ of the relevant pairs; and Corollary 4.3 shows that their average height is at most $(p+1)/2$.

The proof is organized around the CRT data of Section 2. For each integer $k$, the representatives $a_k,b_k,c_k$ define $F_k=a_k/p+b_k/q+c_k/r-k/(pqr)$, which lies in $\{0,1,2\}$ in the range used. Lemmas 2.2 and 2.3 control first and mixed finite differences of $F_k$; Lemma 3.1 then expresses $a_{pqr}(n)$ in three equivalent ways by counting the occurrences of $0$, $1$, or $2$ among translated $F$-values. The proof of Theorem 1.3 in Section 3 classifies the only contributing quadruples $(F_k,F_{k-q},F_{k-r},F_{k-q-r})$ and counts their possible $a_k$-ranges, producing (3.1)--(3.3). Thus the argument is specific to squarefree orders with exactly three prime factors and does not claim a uniform bound independent of the least prime outside the displayed congruence families.

Theorem 1.5 is the jump-one result $|a_{pqr}(n)-a_{pqr}(n-1)|\leq1$. Its proof in Section 5 is not merely an application of the height bound. Lemma 5.1 telescopes the counting formulas of Lemma 3.1 to write the jump as $\tfrac12(N_--N_+)$, where $N_+$ and $N_-$ count the entries equal to $1$ in two four-term collections of translated $F$-values; it also gives parallel formulas using the counts of $0$ or $2$. The first formula gives an a priori bound of $2$. Equality would force one four-term collection to consist entirely of $1$'s and the other to contain no $1$'s. The alternative count formulas then force, after permuting $p,q,r$, a mixed second difference of $F$ to have absolute value $2$, contradicting the values $0,\pm1$ prescribed by Lemma 2.3. This excludes jumps of size $2$ and proves Theorem 1.5.