A technique to study the correlation measures of binary sequences

Abstract

Let E N = ( e 1 , e 2 , … , e N ) be a binary sequence with e i ∈ { + 1 , − 1 } . For 2 ≤ k ≤ N , the correlation measure of order k of the sequence is defined by Mauduit and Sárközy as C k ( E N ) = max M , d 1 , … , d k | ∑ n = 1 M e n + d 1 e n + d 2 … e n + d k | where the maximum is taken over all M ≥ 1 and 0 ≤ d 1 < d 2 < … < d k such that M + d k ≤ N . These measures have been extensively studied over the last decade. Several inequalities for these measures (that hold for all E N for all large enough N ) have been proved, and others conjectured. Further, these measures have been estimated for various special sequences E N . Fix M ≥ 1 and L ≥ 1 . For 1 ≤ a ≤ L , let E M [ a ] = ( e 1 [ a ] , … , e M [ a ] ) be a binary sequence with e i [ a ] ∈ { + 1 , − 1 } . For 2 ≤ k ≤ L we define the correlation measure of order k of the family of sequences E M [ 1 : L ] = { E M [ 1 ] , … , E M [ L ] } as C k ( E M [ 1 : L ] ) = max 1 ≤ a 1 < a 2 < ⋯ < a k ≤ L | ∑ i = 1 M e i [ a 1 ] e i [ a 2 ] … e i [ a k ] | . We use these new correlation measures as a vehicle to study the correlation measures introduced by Mauduit and Sárközy. Alon, Kohayakawa, Mauduit, Moreira, and Rödl recently proved that for each k ≥ 1 there is an absolute constant c 2 k > 0 such that C 2 k ( E N ) ≥ c 2 k N for all E N for all large enough N . thus answering a question of Cassaigne, Mauduit, and Sárközy (in stronger form than an earlier result of Kohayakawa, Mauduit, Moreira, and Rödl). We prove a lower bound on the even correlation measures C 2 k ( E M [ 1 : L ] ) when L > k ( 2 k − 1 ) M and use it to provide an alternate proof of this result. The constant c 2 k in our proof is better than that of Alon, Kohayakawa, Mauduit, Moreira, and Rödl for k = 1 , but poorer for all k ≥ 2 . We study C 3 ( E N ) via C 3 ( E M [ 1 : L ] ) . This allows us to strengthen a recent result of Gyarmati which relates C 3 ( E N ) and C 2 ( E N ) . We prove that given any κ > 0 there is an associated c > 0 (depending only on κ ) such that, for all sufficiently large N , if C 2 ( E N ) ≤ κ N 2 / 3 we have C 3 ( E N ) ≥ c N . This also answers a question of Gyarmati. Finally, the study of C 3 ( E M [ 1 : L ] ) allows us to verify a conjecture of Mauduit. We prove that there is an absolute constant c > 0 such that C 2 ( E N ) C 3 ( E N ) ≥ c N for all E N for all large enough N .