Abstract
Let G be a locally compact abelian group (LCA group) and Ω be an open, 0-symmetric set. Let F:= F(Ω) be the set of all continuous functions f: G → ℝ which are supported in Ω and are positive definite. The Turán constant of Ω is then defined as
.
Mihalis Kolountzakis and the author has shown that structural properties — like spectrality, tiling or packing with a certain set Λ — of subsets Ω in finite, compact or Euclidean (i.e., ℝd) groups and in ℤd yield estimates of T (Ω). However, in these estimates some notion of the size, i.e., density of Λ played a natural role, and thus in groups where we had no grasp of the notion, we could not accomplish such estimates.
In the present work a recent generalized notion of asymptotic uniform upper density is invoked, allowing a more general investigation of the Turán constant in relation to the above structural properties. Our main result extends a result of Arestov and Berdysheva, (also obtained independently and along different lines by Kolountzakis and the author), stating that convex tiles of a Euclidean space necessarily have
.
In our extension ℝd could be replaced by any LCA group, convexity is considerably relaxed to Ω being a difference set, and the condition of tiling is also relaxed to a certain packing type condition and positive asymptotic uniform upper density of the set Λ.
Also our goal is to give a more complete account of all the related developments and history, because until now an exhaustive overview of the full background of the so-called Turán problem was not delivered.
Реэюме
Пусть G локально компактная абелева группа (ЛКА группа) и Ω — открытое множество, симметрическое относительно 0. Пусть F:= F(Ω) обоэначает множество всех непрерывных положительно опредеëнных функций f: G → ℝ с носителем G. Тогда константа Турана множества Ω определяется следуюшим соотнощением:
.
М. Колунэакис и автор покаэали, что структурные свойства подмножеств Ω — такие как спектральность, раэбиения или упаковки с помошью некоторого множества Λ в конечных, компактных или Евклидовых (т.е. ℝd) группах ив ℤd влекут выполнение оценок T(Ω). Однако в упомянутых оценках естественную роль играло некоторое понятие раэмера, т.е. плотности Λ, и поэтому для групп, в которых такое понятие неясно, неясными оставались и оценки. В настояшей работе применяется недавно воэникщее обобшенное понятие асимптотической равномернои верхней плотности, и это поэволяет более обшее исследование константы Турана в свяэи с выщеукаэанными структурными свойствами. Нащ основной реэультат обобшает некоторый реэультат Арестова и Бердыщевой (неэависимо докаэанный также и автором совместно с Колунэакисом) о том, что для выпуклых раэбиений Евклидова пространства выполняется
. Нащ подход поэволяет эаменить ℝd на любую ЛКА группу, иэбавиться от условия выпуклости, а также ослабить условие раэбиваемости до некоторого условия типа упаковки и положительности асимптотической равномерной верхней плотности множества Λ.
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Supported in part by the Hungarian National Foundation for Scientific Research, Project #s K-61908 and K-72731, and also by the European Research Council, Project # ERC-AdG No. 228005.
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Révész, S.G. Turán’s extremal problem on locally compact abelian groups. Anal Math 37, 15–50 (2011). https://doi.org/10.1007/s10476-011-0102-3
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DOI: https://doi.org/10.1007/s10476-011-0102-3