Lacunary sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:51:49Z http://mathoverflow.net/feeds/question/103014 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103014/lacunary-sequence Lacunary sequence Charles 2012-07-24T15:38:18Z 2012-07-25T01:52:39Z <p>Is there a standard definition for a lacunary sequence?</p> <p>Suppose $0 &lt; a_1 &lt; a_2 &lt; \cdots.$</p> <p>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 $$\lim_na_{n+1}-a_n=+\infty.$$</p> <p>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.</p> <p>Further, the <a href="http://www.encyclopediaofmath.org/index.php/Lacunary_sequence" rel="nofollow">EOM entry for lacunary sequence</a> is different from both (a finite form of the first): $$\frac{a_{n+1}}{a_n}\ge\lambda>1.$$</p> http://mathoverflow.net/questions/103014/lacunary-sequence/103019#103019 Answer by quid for Lacunary sequence quid 2012-07-24T16:30:11Z 2012-07-24T16:30:11Z <p>I believe historically the 'lacunary' terminology derives from Hadamard (around <a href="http://mathworld.wolfram.com/Ostrowski-HadamardGapTheorem.html" rel="nofollow">Ostrowski--Hadarmard's Gap Theorem</a>), and there are other results like this, where one needs the condition on the ratio (as opposed to the difference). </p> <p>[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].</p> <p>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.</p> <p>Now, if the lim in the question was meant that the limit has to <em>exist</em> this would seem surprising to me. </p> <p>I never saw the condition imposed for differences as described in the question (but this does not mean that much).</p>