Ongoing trouble with I2C on Atmega328p
I don't see a start bit.  The start bit is when SDA goes low while SCL remains high.  I think every new transaction begins with a start bit.  See P13/14 of the datasheet.  Maybe it's not shown (and off-left of the scope screen)?

Also. perhaps part of what dannyf said.  Try address 0xE0.
The I2C Address of your chip is indeed 0x70.  However this is only 7 bits wide.  Depending on the firmware implementation, one may need to left-shift this value by one, and or-in the read/write bit.

By your scope traces, it looks to me that you're sending and unshifted 0x70, and the result is being nak'd (no chip at that address).

To decipher the transmitted data in the scope traces, look at where the scl goes high, and the condition of sda (is it high or low).

To read from this chip, send 0xE1, then whatever (if any) other data.
To write from this chip, send 0xE0, then whatever to write.

This is where having a logic probe really, really helps.  There are many cheap (and not cheap) tools available.