A Cantor-Lebesgue theorem with variable ``coefficients''

If $\{\phi _n\}$ is a lacunary sequence of integers, and if for each $n$, $% c_n(x)$ and $c_{-n}(x)$ are trigonometric polynomials of degree $n,$ then $% \{c_n(x)\}$ must tend to zero for almost every $x$ whenever $% \{c_n(x)e^{i\phi _nx}+c_{-n}(-x)e^{-i\phi _nx}\}$ does. We conjecture that a similar result ought to hold even when the sequence $\{\phi _n\}$ has much slower growth. However, there is a sequence of integers $\{n_j\}$ and trigonometric polynomials $\{P_j\}$ such that $\{e^{in_jx}-P_j(x)\}$ tends to zero everywhere, even though the degree of $P_j$ does not exceed $n_j-j$ for each $j$. The sequence of trigonometric polynomials $\{\sqrt {n}\sin ^{2n}\frac x2\}$ tends to zero for almost every $x$, although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree $n$ with largest Fourier coefficient equal to $1$, the smallest one ``at'' $x=0$ is $% 4^n\binom {2n}n^{-1}\sin ^{2n}\left ( \frac x2\right ) ,$ while the smallest one ``near'' $x=0$ is unknown.