I wonder if a set of integers $A$ given by an increasing sequence of integers $A=(a_k)_k$ with positive upper density $\limsup_n\frac{\sharp A\cap [0,n]}{n}>0$ satisfies the following property.

For any $\epsilon>0$ there exists an integer $L>0$ such that we can find arbitrarily large intervals $J=[a_p,a_q]$ for which the union $J_L$ of intervals [a_k,a_{k+1}] with $a_{k+1}-a_k>L$ lying inside $J$ represents a proportion less than $\epsilon$ in $J$ :

$$\frac{|J_L|}{|J|}<\epsilon.$$

The example of not piecewise syndetic set with positive upper density that i have in mind satisfies this property. This is certainly false but i have no idea about a counterexample.

I wonder if a set of integers $A$ given by an increasing sequence of integers $A=(a_k)_k$ with positive upper density $\limsup_n\frac{\sharp A\cap [0,n]}{n}>0$ satisfies the following property.

For any $\epsilon>0$ there exists an integer $L>0$ such that we can find arbitrarily large intervals $J=[a_p,a_q]$ for which the union $J_L$ of intervals $[a_k,a_{k+1}]$ with $a_{k+1}-a_k>L$ lying inside $J$ represents a proportion less than $\epsilon$ in $J$ :

$$\frac{|J_L|}{|J|}<\epsilon.$$

The example of not piecewise syndetic set with positive upper density that I have in mind satisfies this property. This is certainly false but I have no idea about a counterexample.