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The phrase "a smaller slice of" is correct and usable in written English.
It can be used to describe a reduced portion or share of something, often in a metaphorical sense.
Example: "In the latest budget proposal, our department is expected to receive a smaller slice of the funding compared to last year."
Alternatives: "a reduced portion of" or "a lesser share of".
Exact(39)
The Shiites for decades got a smaller slice of the Lebanese pie than their numbers merited.
Lotinga denied authors were getting a smaller slice of the publishing cake.
Since the industrialisation of food production, food has taken up a smaller slice of our budgets and our time.
The Polancos may have a smaller slice of the firm, but the pie should start to look a lot bigger.
If a smaller slice of their retirement savings were bouncing around, it might have been easier to remain invested.
"Europe may get a smaller slice of the cake," he said, "due to the rise of the euro, but the overall size of the cake is bigger".
Similar(21)
And I said: "This was always a small slice-of-life piece -- maybe it's old-fashioned".
Solar still provides a small slice of electricity generation.
The sea is a small slice of it.
But marathons have been a small slice of his running.
Or at least a small slice of same.
More suggestions(17)
a smaller segment of
a faster slice of
a smaller fraction of
a narrower slice of
a smaller proportion of
a smaller chunk of
a smaller tranche of
a smaller ring of
a smaller cut of
a smaller share of
a wider slice of
a smaller piece of
a smaller bite of
a smaller number of
a smaller majority of
a smaller group of
a smaller pool of
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com