2 +inf

Is there a standard definition for a lacunary sequence?

Suppose $0 < a_1 < a_2 < \cdots.$

I've read two papers using the term recently. One requires $$\lim_n\frac{a_{n+1}}{a_n}>1 liminf_n\frac{a_{n+1}}{a_n}>1$$ while the other only requires $$\lim_na_{n+1}-a_n=+\infty.$$

The two differ, of course: $a_1=1,\ a_{n+1}=a_n+\sqrt{a_n}$ has a ratio that tends to 1 but a difference that diverges.

Further, the EOM entry for lacunary sequence is different from both (a finite form of the first): $$\frac{a_{n+1}}{a_n}\ge\lambda>1.$$

1

# Lacunary sequence

Is there a standard definition for a lacunary sequence?

Suppose $0 < a_1 < a_2 < \cdots.$

I've read two papers using the term recently. One requires $$\lim_n\frac{a_{n+1}}{a_n}>1$$ while the other only requires $$\lim_na_{n+1}-a_n=+\infty.$$

The two differ, of course: $a_1=1,\ a_{n+1}=a_n+\sqrt{a_n}$ has a ratio that tends to 1 but a difference that diverges.

Further, the EOM entry for lacunary sequence is different from both (a finite form of the first): $$\frac{a_{n+1}}{a_n}\ge\lambda>1.$$