Average Mahler’s measure and $L_p$ norms of Littlewood polynomials
Littlewood polynomials are polynomials with each of their coefficients in the set {-1,1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the 𝐿_{𝑝} norms of Littlewood polynomials of degree 𝑛-1. We show that the arithmetic mea...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
American Mathematical Society
2014-10-01
|
| Series: | Proceedings of the American Mathematical Society, Series B |
| Subjects: | |
| Online Access: | http://www.ams.org/bproc/2014-01-10/S2330-1511-2014-00013-4/ |
| _version_ | 1818750493025894400 |
|---|---|
| author | Stephen Choi Tamás Erdélyi |
| author_facet | Stephen Choi Tamás Erdélyi |
| author_sort | Stephen Choi |
| collection | DOAJ |
| description | Littlewood polynomials are polynomials with
each of their coefficients in the set {-1,1}. We compute
asymptotic formulas for the arithmetic mean values of the
Mahler’s measure and the 𝐿_{𝑝} norms of Littlewood polynomials
of degree 𝑛-1. We show that the arithmetic means of the Mahler’s
measure and the 𝐿_{𝑝} norms of Littlewood polynomials of degree
𝑛-1 are asymptotically 𝑒^{-𝛾/2}√𝑛 and Γ(1+𝑝/2)^{1/𝑝}√𝑛,
respectively, as 𝑛 grows large. Here 𝛾 is Euler’s constant. We
also compute asymptotic formulas for the power means 𝑀_{𝛼} of
the 𝐿_{𝑝} norms of Littlewood polynomials of degree 𝑛-1 for any
𝑝>0 and 𝛼>0. We are able to compute asymptotic formulas for the
geometric means of the Mahler’s measure of the “truncated”
Littlewood polynomials 𝑓 defined by 𝑓(𝑧):=min{|𝑓(𝑧)|,1/𝑛}
associated with Littlewood polynomials 𝑓 of degree 𝑛-1. These
“truncated” Littlewood polynomials have the same limiting
distribution functions as the Littlewood polynomials. Analogous
results for the unimodular polynomials, i.e., with complex
coefficients of modulus 1, were proved before. Our results for
Littlewood polynomials were expected for a long time but looked
beyond reach, as a result of Fielding known for means of
unimodular polynomials was not available for means of Littlewood
polynomials. |
| first_indexed | 2024-12-18T04:20:33Z |
| format | Article |
| id | doaj.art-acbd3b40c4a64ed097c119bdfce5f5da |
| institution | Directory Open Access Journal |
| issn | 2330-1511 |
| language | English |
| last_indexed | 2024-12-18T04:20:33Z |
| publishDate | 2014-10-01 |
| publisher | American Mathematical Society |
| record_format | Article |
| series | Proceedings of the American Mathematical Society, Series B |
| spelling | doaj.art-acbd3b40c4a64ed097c119bdfce5f5da2022-12-21T21:21:14ZengAmerican Mathematical SocietyProceedings of the American Mathematical Society, Series B2330-15112014-10-0111010512010.1090/S2330-1511-2014-00013-4bproc00013Average Mahler’s measure and $L_p$ norms of Littlewood polynomialsStephen Choi0Tamás Erdélyi1 Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada Department of Mathematics, Texas A&M University, College Station, Texas 77842 Littlewood polynomials are polynomials with each of their coefficients in the set {-1,1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the 𝐿_{𝑝} norms of Littlewood polynomials of degree 𝑛-1. We show that the arithmetic means of the Mahler’s measure and the 𝐿_{𝑝} norms of Littlewood polynomials of degree 𝑛-1 are asymptotically 𝑒^{-𝛾/2}√𝑛 and Γ(1+𝑝/2)^{1/𝑝}√𝑛, respectively, as 𝑛 grows large. Here 𝛾 is Euler’s constant. We also compute asymptotic formulas for the power means 𝑀_{𝛼} of the 𝐿_{𝑝} norms of Littlewood polynomials of degree 𝑛-1 for any 𝑝>0 and 𝛼>0. We are able to compute asymptotic formulas for the geometric means of the Mahler’s measure of the “truncated” Littlewood polynomials 𝑓 defined by 𝑓(𝑧):=min{|𝑓(𝑧)|,1/𝑛} associated with Littlewood polynomials 𝑓 of degree 𝑛-1. These “truncated” Littlewood polynomials have the same limiting distribution functions as the Littlewood polynomials. Analogous results for the unimodular polynomials, i.e., with complex coefficients of modulus 1, were proved before. Our results for Littlewood polynomials were expected for a long time but looked beyond reach, as a result of Fielding known for means of unimodular polynomials was not available for means of Littlewood polynomials.http://www.ams.org/bproc/2014-01-10/S2330-1511-2014-00013-4/Mean Mahler’s measuremean 𝐿_{𝑝} normunimodular polynomialLittlewood polynomialMahler’s problem. |
| spellingShingle | Stephen Choi Tamás Erdélyi Average Mahler’s measure and $L_p$ norms of Littlewood polynomials Proceedings of the American Mathematical Society, Series B Mean Mahler’s measure mean 𝐿_{𝑝} norm unimodular polynomial Littlewood polynomial Mahler’s problem. |
| title | Average Mahler’s measure and $L_p$ norms of Littlewood
polynomials |
| title_full | Average Mahler’s measure and $L_p$ norms of Littlewood
polynomials |
| title_fullStr | Average Mahler’s measure and $L_p$ norms of Littlewood
polynomials |
| title_full_unstemmed | Average Mahler’s measure and $L_p$ norms of Littlewood
polynomials |
| title_short | Average Mahler’s measure and $L_p$ norms of Littlewood
polynomials |
| title_sort | average mahler s measure and l p norms of littlewood polynomials |
| topic | Mean Mahler’s measure mean 𝐿_{𝑝} norm unimodular polynomial Littlewood polynomial Mahler’s problem. |
| url | http://www.ams.org/bproc/2014-01-10/S2330-1511-2014-00013-4/ |
| work_keys_str_mv | AT stephenchoi averagemahlersmeasureandlpnormsoflittlewoodpolynomials AT tamaserdelyi averagemahlersmeasureandlpnormsoflittlewoodpolynomials |