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

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If there is a standard definition, it is clearly not well-known enough to be used without stating it; I think I've seen similar things with lim inf instead of lim. For a paper, it's safest to define exactly what you mean. – Zen Harper Jul 24 '12 at 16:12
I think this is a self-answering question. – Boris Bukh Jul 24 '12 at 16:12
@Zen Harper: Certainly the term should not be used without definition! But if the standard definition was A and I needed B, then I shouldn't use the term at all (except to clarify). Thus the question. – Charles Jul 24 '12 at 16:39
In subsequence ergodic theory, the condition is multiplicative, as everywhere else by the sound of it. We don't use any condition on the existence of the limit of the ratios. $n!$ is lacunary for example. – Anthony Quas Jul 24 '12 at 17:50
I've always met this adjective in definitions given locally inside a theorem or inside a paper ("let's define a power series lacunary iff..."). It seems to me a case where it is more useful not to choose a standard definition once and for all. I'd rather leave the freedom to cover time by time each of the various possible cases of set of integers with "large gaps". – Pietro Majer Jul 25 '12 at 8:26

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).

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The difference condition I saw in Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, 2002. Theorem 3.2. I'll know to avoid using that definition myself if the other one is standard (as seems to be the case, more or less). – Charles Jul 24 '12 at 16:37
Oh -- good catch about the limit, I fixed that. – Charles Jul 25 '12 at 1:52
@Charles: thanks for the reference to the paper. – quid Jul 25 '12 at 10:34