Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1<s_{1}<s_{2}<\dots <s_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be positive integers with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gcd (s_{1},s_{2},\ldots ,s_k)=1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Recently, Yu, Chen, and Chen proved that every sufficiently large integer n can be expressed as a finite sum of distinct terms taken from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left\{ s_{1}^{x_{1}}\cdots s_{k}^{x_{k}}:x_{i}\in {\mathbb Z}_{\ge 0}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <msubsup> <mi>s</mi> <mrow> <mn>1</mn> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </msubsup> <mo>⋯</mo> <msubsup> <mi>s</mi> <mrow> <mi>k</mi> </mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> </msubsup> <mo>:</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mfenced> </math></EquationSource> </InlineEquation>. For any real number <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>, we show that there exists an absolute constant c (depending only on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$s_{1},s_{2},\ldots ,s_{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>) such that every sufficiently large integer n can be represented in this form with each term exceeding <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\frac{cn}{(\log n)^{\log k/ \log 2+\varepsilon }}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi mathvariant="italic">cn</mi> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>log</mo> <mi>k</mi> <mo stretchy="false">/</mo> <mo>log</mo> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mfrac> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This extends a 2024 result of Yu.

The Birch–Erdős theorem on exponential type sequences

期刊论文
发表日期：

2026年3月18日

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Honghu Liu

摘要

Let $k\ge 2$ $k \geq 2$ , and let $1<s_{1}<s_{2}<\dots <s_{k}$ $1 < s_{1} < s_{2} < \dots < s_{k}$ be positive integers with $\gcd (s_{1},s_{2},\ldots ,s_k)=1.$ $gcd (s_{1}, s_{2}, \dots, s_{k}) = 1 .$ Recently, Yu, Chen, and Chen proved that every sufficiently large integer n can be expressed as a finite sum of distinct terms taken from $\left\{ s_{1}^{x_{1}}\cdots s_{k}^{x_{k}}:x_{i}\in {\mathbb Z}_{\ge 0}\right\} $ $\{s_{1}^{x_{1}} \dots s_{k}^{x_{k}} : x_{i} \in Z_{\geq 0}\}$ . For any real number $\varepsilon $ $ε$ , we show that there exists an absolute constant c (depending only on $\varepsilon $ $ε$ and $s_{1},s_{2},\ldots ,s_{k}$ $s_{1}, s_{2}, \dots, s_{k}$ ) such that every sufficiently large integer n can be represented in this form with each term exceeding $\frac{cn}{(\log n)^{\log k/ \log 2+\varepsilon }}.$ $\frac{cn}{{(log n)}^{log k / log 2 + ε}} .$ This extends a 2024 result of Yu.