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<p>The paper deals with some problems related to sums of the type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sum _{k=0}^{Q-1} f(k\theta + \varphi )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>Q</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>θ</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where <i>f</i> is a continuous function and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> is an irrational number. We study the limit behavior of these sums as <i>Q</i> tends to infinity. Our results depend on the class to which function <i>f</i> belongs, as well as on Diophantine properties of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>.</p>

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Liminf-results for sums along the Kronecker sequences

  • 期刊论文

  • 发表日期：

    2026年3月30日

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  • Artem Chebotarenko

摘要

The paper deals with some problems related to sums of the type \(\sum _{k=0}^{Q-1} f(k\theta + \varphi )\) ∑ k = 0 Q - 1 f ( k θ + φ ) where f is a continuous function and \(\theta \) θ is an irrational number. We study the limit behavior of these sums as Q tends to infinity. Our results depend on the class to which function f belongs, as well as on Diophantine properties of \(\theta \) θ .