Random trigonometric polynomials have smaller average Nikolskii factors

What the study found: The study found that the average Nikolskii factor for random trigonometric polynomials has exact orders that are smaller than the worst-case bounds. For 1 ≤ p < q < ∞, the average factor is of order n^0, meaning it stays constant. For 1 ≤ p < q = ∞, it is of order (ln n)^{1/2}.
Why the authors say this matters: The authors state that these average-case orders contrast with the larger worst-case bounds, which are n^{1/p-1/q} for q < ∞ and n^{1/p} for q = ∞. They also say the results generalize to random multivariate trigonometric polynomials.
What the researchers tested: The researchers studied trigonometric polynomials with independent Gaussian coefficients, specifically N(0, σ^2) coefficients. They examined the Nikolskii factor, defined as the ratio of the q-norm to the p-norm of the polynomial.
What worked and what didn't: The average Nikolskii factor was constant in the case 1 ≤ p < q < ∞. In the case 1 ≤ p < q = ∞, it grew like the square root of log n. The abstract does not report failures or negative results beyond comparison with worst-case bounds.
What to keep in mind: The summary available here does not describe proof details, assumptions beyond Gaussian independence, or other limitations. The abstract only states the exact orders and the multivariate generalization.

Key points

• The paper studies average Nikolskii factors for random trigonometric polynomials.
• For 1 ≤ p < q < ∞, the average factor is constant in n.
• For 1 ≤ p < q = ∞, the average factor grows like (ln n)^{1/2}.
• These average-case orders are smaller than the stated worst-case bounds.
• The abstract says the result extends to random multivariate trigonometric polynomials.