6 problems

Exponent-two conjecture. The exponent $2$ should be the optimal exponent in a quantitative estimate in terms of the Fraenkel asymmetry, instead of the exponent $3$ in the displayed…

Let $n$ be a positive integer and let $\alpha_1,\ldots,\alpha_{2n}$ be real numbers satisfying … Consider the alternating exponential sum $\sum_{j=1}^{2n}(-1)^j e^{i\alpha_jt}$. Al…

Characteristic-function rearrangement conjecture. For every measurable $A$ with $|A|<\infty$ and every $W>0$,

Let $T,W>0$. For a function $f\in L^2(\mathbb{R})$ with $\|f\|_2=1$ and $|\operatorname{supp}f|\leq T$, consider its quadratic spectral concentration … Here…

Let $Q$ be a $d$-dimensional cube and let $f^*$ denote the monotone rearrangement of $f$ on $[0,1]$. Let $c(d)$ be the constant appearing in the dimension-reduction conjecture for…

Let $d\geq 1$ be an integer, let $\Sigma\subset\mathbb{R}^d$ be a set of finite positive measure, and let $\Sigma^*$ denote the centered ball with $|\Sigma^*|=|\Sigma|$. For…