Ensembles de non synthèse pour certains poids dissymétriques sur la droite

Let w be a weight and let ${\displaystyle L^1(ℝ,w)}$ be the algebra of all measurable functions f on ℝ such that fw is integrable. It is known that if S is a closed countable subset of ℝ then S satisfies the spectral synthesis in ${\displaystyle L^1(ℝ,w)}$ for all weights w such that ${\displaystyle \{w\left(t\right)=1f\text{or}t\ge 0,lim{\supset }_{t\to \infty }\frac{\mathrm{Log}w(-t)}{t^{\frac{1}{2}}}=0}$. We prove here that this result fails for a large class of uncountable closed subsets of ℝ with Lebesgue measure zero.