Growth of the product ${\displaystyle {\prod }^n\_\{j=1\}(1-x^{a_j})}$

We estimate the maximum of ${\displaystyle {\prod }^n\_\{j=1\}|1-x^{a_j}|}$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when ${\displaystyle a_j}$ is ${\displaystyle j^k}$ or when ${\displaystyle a_j}$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when ${\displaystyle a_j}$ is j.    In contrast we show, under fairly general conditions, that the maximum is less than ${\displaystyle \frac{2^n}{n^r}}$, where r is an arbitrary positive number. One consequence is that the number of partitions of m with an even number of parts chosen from ${\displaystyle a₁,...,a_n}$ is asymptotically equal to the number of such partitions with an odd number of parts when ${\displaystyle a_i}$ satisfies these general conditions.