Lacunary sequence

2012-07-24T15:38:18Z

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 $$ \liminf_n\frac{a_{n+1}}{a_n}>1 $$ 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): $$ \frac{a_{n+1}}{a_n}\ge\lambda>1. $$

Answer by quid for Lacunary sequence

2012-07-24T16:30:11Z

I believe historically the 'lacunary' terminology derives from Hadamard (around Ostrowski--Hadarmard's Gap Theorem), and there are other results like this, where one needs the condition on the ratio (as opposed to the difference).

[Indeed in some papers dealing with lacunary sequences one can read things like: let be a lacunary sequences that is one fulfilling the Hadamard Gap condition; meaning that the ratio of succesive terms is bounded away from one].

Whether one imposes this for all as OEM or only asymptotically as the first one in the question (admitting that lim is actually meant as lim inf) is for these applications not really relevant. And roughly speaking should not be of relevance too frequently for question one typically asks on inifnite sequences, series and alike.

Now, if the lim in the question was meant that the limit has to exist this would seem surprising to me.

I never saw the condition imposed for differences as described in the question (but this does not mean that much).

Lacunary sequence - MathOverflow

Lacunary sequence

Answer by quid for Lacunary sequence