In this paper, we estimate the minimal cardinality of the transcendental dilated sumset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S+\mu \cdot T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>+</mo> <mi>μ</mi> <mo>·</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>, where S and T are non-empty finite subsets of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>-Module <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a transcendental number. Additionally, we optimize the lower bounds for the cardinalities of the sets <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S+NT$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>+</mo> <mi>N</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S+N\, \hat{/phantom{i}}\, T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>+</mo> <mi>N</mi> <mspace width="0.166667em" /> <mover accent="true"> <mrow> <mo stretchy="false">/</mo> <mi>p</mi> <mi>h</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> <mi>o</mi> <mi>m</mi> <mi>i</mi> </mrow> <mo stretchy="false">^</mo> </mover> <mspace width="0.166667em" /> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>, where N, S, and T are finite subsets of positive elements of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>-Module <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>.

On folded and transcendental dilated sumsets

期刊论文
发表日期：

2026年4月9日

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Madhav Sharma,
Sandeep Singh

摘要

In this paper, we estimate the minimal cardinality of the transcendental dilated sumset $S+\mu \cdot T$ $S + μ \cdot T$ , where S and T are non-empty finite subsets of $\mathbb {R}$ $R$ -Module $\mathbb {R}$ $R$ and $\mu $ $μ$ is a transcendental number. Additionally, we optimize the lower bounds for the cardinalities of the sets $S+NT$ $S + N T$ and $S+N\, \hat{/phantom{i}}\, T$ $S + N \hat{/ p h a n t o m i} T$ , where N, S, and T are finite subsets of positive elements of $\mathbb {Z}$ $Z$ -Module $\mathbb {Z}$ $Z$ .